GREEN’S FUNCTIONS FOR VARIANTS OF THE
SCHRAMM-LOEWNER EVOLUTION
By
Benjamin Mackey

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mathematics—Doctor of Philosophy
2017

ABSTRACT
GREEN’S FUNCTIONS FOR VARIANTS OF THE SCHRAMM-LOEWNER
EVOLUTION
By
Benjamin Mackey
We prove upper bounds for the probability that a radial SLEκ curve comes within specified radii of n different points in the unit disc. Using this estimate, we then prove a similar
upper bound for the probability that a whole-plane SLEκ passes near any n points in the
complex plane. We then use these estimates to show that the lower Minkowski content of
both the radial and whole-plane SLEκ traces has finite moments of any order.
For κ ≤ 4, the reverse flow of the Loewner equation driven by

√

κBt generates a random

continuous function φ : R+ → R+ called the conformal welding. In studying backward
SLE, this plays the roll of the global random object, rather than the SLE trace. Given
any x, y > 0 we use the Girsanov theorem to construct a family of probability measures,
depending on some parameters, under which the conformal welding satisfies φ(x) = y almost
surely. For one such law, we prove a one-point estimate for the backward SLE welding and
show how it coincides with the Green’s function. In another case, we decompose the law of
the welding conditioned to pass through (x, y) into two pieces. Using this decomposition,
we integrate this law over a set U ⊂ [0, ∞) × [0, ∞) to get a new measure on weldings which
is absolutely continuous with respect to the original backward SLE welding. Moreover, the
Radon-Nikodym derivative is given by the capacity time that the graph of φ spends in U .
In the last chapter, we study a generalization of the chordal Loewner equation called
chordal measure driven Loewner evolution. We show existence of a solution to the equation,
and a one-to-one correspondence between the appropriate measures and all continuously

growing families of H-hulls. In [19], the notion of measure driven Loewner evolution was
first introduced in the radial setting, and a similar theorem was proven. This result is pure
complex analysis, without any reference to probability theory.

I would like to dedicate this thesis to my wonderful wife, Lauren Conway, who has never
waivered in her support of and confidence in me. I could not have written this without her
standing by my side.“Best of wives and best of women.”

iv

ACKNOWLEDGMENTS

I would like to expresss my sincerest gratitude for my advisor, professor Dapeng Zhan, for
introducing me to the subject of SLE and for continuously providing support for my research.
His guidance in selecting research problems on which to work has been invaluable, as has his
perpetual willingness to comment on my progress and to steer my efforts into a productive
direction. I also would like to thank him for his essential assistance in editting my thesis
into its current form.
I would also like to thank the members of my committee, professors Shlomo Levantal,
Ignacio Uriate-Tuero, and Yimin Xiao, for the taking the time to read and comment on
my research. I am also grateful that they have taken the time to serve on my committee
during the summer semester. I particularly want to thank professor Ignacio Uriarte-Tuero
for serving as a constant figure in my education here at Michigan State University and always
expressing an interest in my work. His real analysis course pushed me harder than any other
course I have ever taken or will take, and I cannot quantify how much I gained from the
experience.
Lastly, I would like to thank my friends and family for their constant support during
my time in graduate school. Tyler Bongers, Guillermo Rey, Reshma Menon, Emily Olson,
Rodrigo Matos, Leonardo Abbrescia, Rami Fakhry, Chelsea Law, David Zach, Matt Alexander, Mark Bissler, Isaac Defrain, and many more have worked with me, run with me, rock
climbed with me, or generally kept me sane during my time at Michigan State University.
My wife, parents, brother, and in-laws have also all been incredibly accomidating of my
frequent absences and interstate travel as I work on this dissertation.

v

TABLE OF CONTENTS

Chapter 1 Introduction . . . . . . . . . . . . . .
1.1 Description of SLE . . . . . . . . . . . . . . .
1.2 Definitions of variants of SLE . . . . . . . . .
1.2.1 Hulls in the complex plane . . . . . . .
1.2.2 Chordal Loewner evolution in H . . . .
1.2.3 Radial Loewner evolution in D . . . . .
1.2.4 Whole-plane Loewner evolution in C .
1.2.5 Backward Loewner evolution . . . . . .
1.3 Natural parametrization and Green’s functions
1.4 Generalizations of the Loewner equation . . .

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Chapter 2 Multipoint estimates for radial and whole-plane
2.1 Statement of results . . . . . . . . . . . . . . . . . . . . . . .
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Crosscuts and prime ends . . . . . . . . . . . . . . .
2.2.2 The covering space H∗ for D . . . . . . . . . . . . . .
2.2.3 Extremal length and conformal transformations . . .
2.3 Interior and boundary estimates . . . . . . . . . . . . . . . .
2.4 Components of crosscuts . . . . . . . . . . . . . . . . . . . .
2.5 Concentric circles . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Main theorems . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Technical lemmas . . . . . . . . . . . . . . . . . . . . . . . .

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1
1
4
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6
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9
14

SLE
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17
17
19
19
21
24
27
35
43
50
58

Chapter 3 Decomposition of backward SLE in the capacity parametrization
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Processes with a random lifetime . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Glossary of notation for Ito’s formula calculations . . . . . . . . . . . . . . .
3.4 Generalized Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Capacity parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 One-point estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 4 Measure driven chordal Loewner evolution
4.1 Theorem statement and definitions . . . . . . . . . . .
4.2 Preliminary results . . . . . . . . . . . . . . . . . . . .
4.3 A disintegration lemma . . . . . . . . . . . . . . . . . .
4.4 Elementary calculations . . . . . . . . . . . . . . . . .
4.5 Proof of existence . . . . . . . . . . . . . . . . . . . . .
4.6 Growing Hulls . . . . . . . . . . . . . . . . . . . . . . .

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63
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121

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

vi

Chapter 1
Introduction

1.1

Description of SLE

This thesis is concerned with the study of conformally invariant randomly growing processes
in subdomains of the complex plane. The main object is the Schramm-Loewner evolution,
or SLE, which was introduced by Oded Schramm in [29] as a candidate for several lattice
models in statistical physics. The processes all arise from solving a variant of the Loewner
differential equation driven by time scaled Brownian motion. In all cases, there is a domain
D ⊂ C and a random curve γ contained in the closure of D starting and ending at fixed
points.
To motivate the definition, we will informally describe the loop erased random walk in
a domain from a to b. Let D ⊂ C be a simply connected domain, and let δ > 0 be a small
number. Consider the lattice Dδ = D ∩ δZ2 , and let γδ be the path of a simple random walk
on Dδ starting at the point (nearest to) a ∈ ∂D, and stopped when it hits a point (nearest
to) b in D, the closure of D. The path γδ can be turned into a simple path γδLE by removing
all of the loops. Is there is a random path γ so that γδLE → γ in law as δ → 0? SLE provides
an answer, which is that the scaling limit is an SLE curve.
SLE has a parameter κ > 0 on which the behavior of the SLEκ path depends. For the
loop erased random walk, SLE2 has been proven to be the scaling limit for the loop erased

1

random walk [13]. Several other lattice models have proven to converge to SLEκ for other
values of κ. SLE3 and SLE16/3 are the scaling limits for Ising model interfaces [35], SLE6
is the scaling limit of the critical percolation explorer curve [34], SLE4 is the scaling limit of
the harmonic explorer [30], and SLE8 is the scaling limit of the uniform spanning tree Peano
curve [13].
There are several varieties of SLE which differ based on where they begin and end, and
each variety has a standard domain in which it is easier to define and study. The two primary
types are called chordal SLEκ , which begins and ends at fixed boundary ponts, and radial
SLEκ , which begins at a fixed boundary point and ends at a fixed interior point.
Making this more precise, fix a simply connected domain D, and fix a point a ∈ ∂D
(technically, a should be a prime end of D, but we will not define that now), and let b ∈ D.
If b ∈ ∂D (or b is a prime end of D), then chordal SLEκ in D from a to b is a random path in
D which begins at a and ends at b almost surely. The standard domain for studying chordal
SLE is the upper half-plane H = {z ∈ C : Im(z) > 0}, starting at 0 and ending at ∞, where
∞ can seen as a point in the Riemann sphere. If b is in the interior of D, then radial SLEκ
in D from a to b is a random path in D which begins at a and ends at b almost surely. The
standard domain for radial SLEκ is the unit disc D = {z ∈ C : |z| < 1}, starting at 1 and
ending at 0.
Once the SLE path is defined in its standard domain, in either case, it can be defined
in any simply connected domain D. To define chordal SLEκ in a simply connected domain
D from a to b, let φ : H → D be a conformal tranformation with φ(0) = a and φ(∞) = b.
Then if µκ is the probability measure on paths in H which is the law of the standard chordal
SLEκ , then the law of SLEκ in D from a to b is given by the pushforward meaure induced
by φ on µκ .
2

This describes conformal invarience. Schramm proved that the only processes which have
conformal invarience and the domain Markov property are SLE processes. These properties
are described more precisely as:
• (Conformal invarience) If γ is an SLEκ path in D from a to b, and φ : D → D is a
conformal transformation with φ(a) = a and φ(b) = b , then φ(γ) is a time changed
SLEκ path in D from a to b .
• (Domain Markov property) Let T be a stopping time with respect to the filtration
generated by an SLEκ path in a domain D from a to b. For any t > 0, Let Dt be the
component of D\γ[0, t] which contains the target point b. Then if γ T (t) = γ(T + t),
after conditioning on γ[0, T ], the path γ T is an SLEκ path in DT from γ(T ) to b.
The laws of chordal SLE and radial SLE are not absolutely continuous with each other,
but there is local absolute continuity [31]. If γ is radial SLEκ in D from 1 to 0, let > 0 and
T = inf{t > 0 : |γ(t)| = }, which is finite a.s.. Then radial SLEκ restricted to time t ≤ T
is absolutely continuous with respect to a stopped chordal SLEκ up to a time change, so
the paths share many of the same local properties. For example, in each case, the Hausdorff
dimension of the SLEκ path is d = min{1 + κ/8, 2} [3]. Also, the path has two phase
transitions as κ varies [27].
We state the transitions when γ is the chordal SLEκ path in H from 0 to ∞. For
0 < κ ≤ 4, γ(0, ∞) is a simple path which stays in H. For κ ≥ 8, γ is a space filling path.
That is, γ(0, ∞) = H almost surely. For 4 < κ < 8, γ has both boundary intersections and
self intersections. If κ < 8, for any z ∈ H, we have P[z ∈ γ] = 0. This is proven in [27]
reducing the SLEκ maps started on the boundary to the study of Bessel processes.
Note that if κ ≤ 4, the domain DT , as described in the domain Markov property, is
3

exactly equal to D\γ[0, T ], since the path is simple and has no boundary intersections. For
κ > 4, DT the piece of the domain which has not yet been cut off from the endpoint by the
path.
Another variety of SLE is called whole-plane SLE, which is a path in C which grows from
0 to ∞. It can be seen as a limit of radial SLEκ paths in large discs from the boundary to
ˆ to any other b ∈ C,
ˆ where C
ˆ is
0. This can be extended to whole-plane SLE from any a ∈ C
the extended complex plane, by applying a M¨obius transformation which takes 0 to a and
ˆ
∞ to b, which gives an SLE path connecting any two interior points of C.
The domain Markov property has a slightly different form for whole-plane SLEκ . If γ ∗ is
the whole-plane SLEκ path, then γ ∗ is a path γ ∗ : (−∞, ∞) → C. For any stopping time T ,
the path γ T (t) = γ ∗ (T + t), conditioned on γ ∗ (−∞, T ], is a a radial SLEκ in C\DT , where
DT is the unbounded component of C\γ ∗ (−∞, T ].

1.2

Definitions of variants of SLE

There are several varieties of the Loewner equation. First, we describe the types of hulls
which correspond to each Loewner equation. We then briefly discuss the chordal equation,
then introduce radial and whole-plane Loewner equations. Complete details can be found in
[7]. We also introduce the reverse Loewner equation and define backward SLE.

1.2.1

Hulls in the complex plane

Let H = {z ∈ C : Im(z) > 0} be the upper half-plane. A set K ⊂ H is called an H-hull if K
is relatively compact in H and H\K is simply connected. Then there is a unique conformal
map gK : H\K → H with gK (z) = z +

c
+ O(z −2 ) as z → ∞. The constant c is called
z
4

the half-plane capacity, and is denoted by hcap(K), and it is equal to 0 if and only if K is
empty. We call a domain H ⊂ H and H-domain if H = H\K for some H-hull K.
Half-plane capacity is monotonic, in that if K1 ⊂ K2 are both H-hulls, we have hcap(K1 ) ≤
hcap(K2 ). Moreover, if K1 ⊂ K2 are both H-hulls, we can define a new H-hull K2 /K1 :=
gK1 (K2 \K1 ), called the quotient hull. In [26], it has been shown that hcap(K2 ) = hcap(K1 )+
hcap(K2 /K1 ). Therefore, if K1 is properly contained in K2 , it follows that hcap(K1 ) <
hcap(K2 ). Half-plane capacity also satisfies the scaling rule hcap(rK + a) = r2 hcap(K) for
any a, r ∈ R.
In [26], the authors define the support of an H-hull K and prove the following results.
ˆ = K ∪ K ∪ BK .
Let BK = K ∩ R, and let K = {z : z ∈ K}. Define the double of K by K
ˆ = C\SK , for a compact SK ⊂ R, which we will call the support of K. Then
Then gK (C\K)
−1 defined in H can be extended conformally to C\S by the Schwarz
the inverse fK := gK
K

reflection principle, but no further. If K only has one component, then SK is an interval,
∗ denote the convex hull of S ,
but in general K can consist of several components. Let SK
K
∗ is a compact interval. If K ⊂ K , then S ∗ ⊂ S ∗ .
so that SK
1
2
K1
K2

Let D = {z ∈ C : |z| < 1} be the unit disc. A D-hull is a set K ⊂ D which is relatively
closed in D, 0 ∈
/ K, and D\K is simply connected. By the Riemann mapping theorem, there
is a unique conformal map gK : D\K → D with gK (0) = 0 and gK (0) > 0. The D-capacity
of K is defined by dcap(K) = ln(gK (0)). A set D ⊂ D is called a D-domain if D = D\K for
a D-hull K.
D-capacity can be readily shown to be monotonic with respect to inclusion. If K1 ⊂ K2
and the quotient hull K2 /K1 := gK1 (K2 \K1 ), then gK2 = gK /K ◦ gK1 , and the chain rule
2 1
implies that gK (0) = gK /K (0)gK (0). Taking the natural logarithm of both sides yields
2
1
2 1
the monotonicity.
5

A compact hull in C is a set K ⊂ C such that 0 ∈ K, K is connected and compact, and C\K is connected. Then there is a unique conformal map FK : C\D → C\K
with limz→∞ FK (z)/z > 0. Looking at C\K under the inversion z → 1/z gives a simply connected domain U containing 0, for which there is a unique conformal Riemann map
fK : D → U with fK (0) > 0. Then FK (z) = 1/fK (1/z), and the logarithmic capacity of K
is defined by cap(K) = log(fK (0)).

1.2.2

Chordal Loewner evolution in H

The chordal Loewner equation describes the growth from a boundary point to another boundary point. The standardized domain is the upper half-plane H = {z ∈ C : Im(z) > 0}, and
the chordal Loewner equation driven by a real valued λ ∈ C[0, ∞) at z ∈ H is

∂t gt (z) =

2
,
gt (z) − λt

g0 (z) = z.

(1.1)

If τz = inf{t > 0 : Im(gt (z)) = 0} is the lifetime of (1.1) at z, define Kt = {z ∈ H : τz ≤ t}.
Then Kt is an H-hull with half-plane capacity hcap(Kt ) = 2t, and gt = gKt is the conformal
map sending H\Kt to H.
If κ > 0 and Bt is a standard one dimensional Brownian motion, then (1.1) driven
λt =

√

κBt is called chordal SLEκ . In [27], it is proven that there is a path γ : [0, ∞) → H

called the chordal SLEκ trace, also called the path or curve, which grows from 0 to ∞. Then
for each t > 0, H\Kt is the unbounded component of H\γ[0, t].

6

1.2.3

Radial Loewner evolution in D

Given any real valued continuous function λ ∈ C[0, ∞), the radial Loewner equation driven
by λ at z ∈ D,
∂t gt (z) = gt (z)

eiλ(t) + gt (z)
eiλ(t) − gt (z)

,

g0 (z) = z.

(1.2)

If τz = inf{t > 0 : |gt (z)| = 1} is the lifetime of (1.2) at z ∈ D and Kt = {z ∈ D : τz ≤ t},
then Kt is a D-hull with dcap(Kt ) = t.
For κ > 0, the radial SLEκ process is the solution (1.2) for λ(t) =

√

κBt , where Bt is

standard one dimensional Brownian motion. Similarly to the chordal case, there is a a radial
trace γ : [0, ∞) → D so that D\Kt is the component of D\γ[0, t] containing 0 with γ(0) = 1
and γ(∞) = 0. The radial SLEκ trace has the same phrase transitions as the chordal SLEκ
and the same dimension. The radial SLEκ process can be studied by looking at the covering
space for D under the exponential map eiz , which is the cylinder H∗ = {[z]∼ : z ∈ H}, where
z ∼ w if z − w ∈ 2πZ. This space will be discussed in more detail in Section 2.2.2.

1.2.4

Whole-plane Loewner evolution in C

If λ ∈ C(−∞, ∞) is real valued and z ∈ C\{0}, the whole-plane Loewner equation driven
by λ is
e−iλt + gt∗ (z)
∂t gt∗ (z) = gt∗ (z) −iλ
, and lim et gt∗ (z) = z.
t→−∞
e t − gt∗ (z)

(1.3)

This can be interpreted as a radial Loewner equation driven by −λ started from t = −∞.
For each t ∈ (−∞, ∞), Kt = {z ∈ C : τz ≤ t} is a compact hull in C with logarithmic
capacity cap(Kt ) = t, and gt∗ : C\Kt → C\D conformally.
To define whole-plane SLEκ , we need to define two sided Brownian motion. Let Bt1 and

7

Bt2 be independent standard one dimensional Brownian motions, and let Y be an independent
√
√
uniform [−π/ κ, π/ κ] random variable. Construct a two sided Brownian motion B : R →
2 + Y if t < 0. Then g ∗ is a whole-plane SLE if it
R by Bt = Bt1 + Y if t ≥ 0, and Bt = B−t
κ
t
√
solves (1.3) with λt = κBt .

There is a curve γ ∗ : (−∞, ∞) → C called the whole-plane SLEκ trace. For each
t ∈ R, C\Kt is the unbounded component of C\γ ∗ (−∞, t]. Also, limt→−∞ γ ∗ (t) = 0 and
limt→∞ γ ∗ (t) = ∞, and so the whole-plane SLEκ curve grows from 0 to ∞ in the entire
complex plane. Since at any time s ∈ R the process (Bt )t≥s is a Brownian motion with
a random start time, it follows that γ ∗ has the same phrase transitions and dimension
depending on κ as the chordal and radial processes.

1.2.5

Backward Loewner evolution

The backward Loewner equation started at z ∈ H driven by λt ∈ C[0, T ) is

∂t ft (z) =

−2
,
ft (z) − λt

f0 (z) = z.

(1.4)

The process (ft )t≤T is called the backward Loewner process driven by λ. Suppose solving
equation (1.1) with the function t → λ(t0 − t) generates a forward Loewner trace t →
γt0 (t0 − t) for each t0 ≤ T . Then we say that λ generates a family of backward Loewner
traces (γt0 )t0 <T . If the paths γt0 are all simple paths, then ft0 : H → H\γt0 [0, t0 ] is
conformal for each t0 ≤ T . Note that in this case, growth does not occur at the tip as in the
forward case. Rather, growth occurs at the base of the curve, which is λ(t) at each t, and
the rest of the curve is the image of the previous curves under a conformal map.
Suppose κ > 0, and let λt =

√

κBt . Then the process given by solving (1.4) is called

8

the backward SLEκ process, and will be denoted by BSLEκ . For a fixed T < ∞, the trace
γT − λT has the same law as a forward SLEκ trace on [0, T ]. So, to study the SLE trace at a
finite time, it suffices to study backward SLE, where the Loewner equation may be easier to
work with. However, the backward SLE traces are not a good global object to study, since
the images γT [0, T ] evolve over time, rather than grow from the tip. If the traces are simple,
there is another object called the conformal welding which serves as global object to study.
For κ ≤ 4, the SLEκ traces are simple, and so the BSLEκ process generates a random
function φ : R+ → R+ called the conformal welding. The conformal maps ft can be extended
to the boundary of H. For any x ∈ R, let τx denote the lifetime of the backward Loewner
equation, which is finite almost surely. For x, y > 0, the welding is defined by φ(x) = y if
τx = τ−y , which is true if and only if ft (x) = ft (−y) for some t > 0. Then φ is a random
monotonic function on R+ . This welding is the main object of study in [26], where the
welding is shown to satisfy a reversability property analogous to a fundamental property of
the forward SLE trace [38] [20].
We will also need to keep track of which points are welded together at a given capacity
time t. To this end, let bt = sup{x > 0 : τx ≤ t} and at = sup{y > 0 : τ−y ≤ t}, so that
ft (bt ) = ft (−at ) = λt for every t. Define Φ(t) = (bt , at ), which we will call the welding
curve. Then if Q1 = [0, ∞) × [0, ∞) is the first quadrant, Φ : [0, ∞) → Q1 is a random path
whose image is the graph of the welding function φ.

1.3

Natural parametrization and Green’s functions

The lattice models which converge to SLE paths converge not only as sets, but also as
continuous functions. To make that conclusion, the lattice model paths and the SLE paths

9

must be parametrized in a particular way. By construction, SLE is parametrized so that the
capacity of the path grows as a constant rate. The lattice model paths which converge to
SLE must also use the capacity parametrization, which is not the natural way to run the
model. It is desirable to run the models so that each discrete segment takes the same length
of time. The most intuitive approach is to simulaneously run the SLE path parametrized by
length, but since the fractal dimension of SLEκ is d = min{1 + κ/8, 2} > 1, the length of
the SLEκ trace is always infinite.
Recently, there has been a program to develop a d-dimensional measurement of length
for chordal SLEκ which can be used to study convergence. In [12], the Doob-Meyer theorem
was used to create an increasing process related to the path which was called the natural
parametrization, or natural length, for κ < 5.021.... It was conjectured to coincide with some
d-dimensional measurment. In [17], the Doob-Meyer construction was extended to all κ < 8.
In [22], it was proven that the Hausdorff measure of the path is 0 almost surely. Instead, the
Minkowski content of the path has been proven to be the correct candidate for the natural
parametrization. The d-dimensional Minkowski content of a set E ⊂ C is defined by

Contd (E) = lim rd−2 Area{z ∈ C : dist(z, E) < r},
r→0

provided that the limit exists. In [9], it was proven that the d-dimensional Minkowski content
of the path exists almost surely, and that it differs from the Doob-Meyer construction by a
multiplicative constant. The first proof of convergence in the natural parametrization is [14],
where it is proven that the loop erased random walk converges to SLE2 when parametrized
by the Minkowski content.
One of the main tools for studying the natural parametrization is called the Green’s

10

function, which gives the normalized probability that the SLE path passes through a point.
For κ ≥ 8, P[z ∈ γ] = 1 for all z, so the question is not interesting. For κ ∈ (0, 8),
P[z ∈ γ] = 0 implies that the probability P[dist(z, γ) < r] should decay as r → 0. Moreover,
if the dimension of the path is d, the probability should decay as r2−d . The Green’s function
is defined to be
G(z) = lim rd−2 P [dist(z, γ) < r] ,
r→0

provided this limit exists. In the chordal case, this was first studied with the event {radH\γ (z) <
r} rather than distance, where radD (z) is defined to be the conformal radius of D at z (defined in Section 2.2.2).
In [27], it was observed that for chordal SLEκ and any z ∈ H, the process

Mt (z) = |gt (z)|2−d G(gt (z) −

√

κBt )

should be a local martingale. To construct the natural parametrization in [12], the authors
weighted the probability measure by Mt (z) and used the Girsanov theorem to construct
what is called two-sided radial SLEκ through z, which is chordal SLEκ conditioned to pass
through z. In order to further study the Minkowski content, the two point Green’s function
was proven to exist and analyzed [9] [10] [16]. In [23] and [24], the Green’s function for any
finite number of points was proven to exist for chordal SLEκ . That is,
n

rkd−2 P ∩n
k=1 dist(zk , γ) ≤ r

G(z1 , . . . , zn ) = lim

r→0

k=1

was proven to exist, where r = (r1 , . . . , rn ) and z1 , . . . , zn ∈ H are arbitrary. The upper
bound for the multipoint Green’s function found in [23] is used to prove that the Minkowski

11

content has finite moments of all orders.
In [37], the construction of two-sided radial SLE using the Girsanov theorem is generalized. For ρ ∈ R and z ∈ H, functions Gρ (z) are found so that if

ρ

Mt (z) = |gt (z)|p Gρ (gt (z) −

√
κBt )

ρ

for an appropriate power p = p(ρ), then Mt (z) is a local martingale which can be used to
ρ

ρ

create a measure Pz under which Pz [z ∈ γ] = 1. A few special cases of these generalized
Green’s functions are used to prove decompositions of the SLEκ path. In each case, the
ρ

ρ

measures Pz are averaged over z ∈ U weighted by Gρ (z) to generate a new measure PU ,
which is absolutely continuous with respect to the original measure P. The Radon-Nikodym
derivatives for three of these constructions are found, and are shown to be the Minkowski
content of γ ∩ U , the capacity time spent by γ in U , and the analogue of the natural
parametrization for the boundary intersection of γ in U ⊂ R for κ ∈ (4, 8). The result for
Minkowski content extends the result in [5] which was proven for κ ≤ 4. These decomposition
theorems have been used to define SLE loop measures [41].
The Green’s function for radial SLEκ is not as well understood as the Green’s function
for chordal SLEκ . In [2], the conformal radius version of the one-point Green’s function is
proven to exist for κ ∈ (0, 8), though an exact form is only found for κ = 4. In Chapter 2,
we establish an upper bound for the probability that radial SLEκ in D from 1 to 0 passes
near any finite collection of points in D. This should serve as the first step towards proving
the multipoint Green’s function for radial SLEκ exists. The strategy is to move from the
covering space H∗ of D under the exponential map eiz , where H∗ is the upper half-plane H
under the equivalence relation z ∼ w if z −w ∈ 2πZ. Here, one-point estimates are developed

12

in [2] for both the boundary and interior case, which we adapt into appropriately general and
conformally invarient estimates. After establishing a sufficiently robust one-point estimate
for radial SLEκ , we follow the strategy in [23] to extend the estimate to multiple points.
The Minkowski content of radial SLEκ and whole-plane SLEκ have not yet been proven
to exist, but the lower content

Contd (E) = lim inf rd−2 Area{z ∈ C : dist(z, γ) ≤ r}
r→0

always exists. Using the multipoint estimate for radial SLEκ we develop, we prove that
Contd (γ) has finite moments of all orders. If the Minkowski content were proven to exist,
the same proof would show that Contd (γ) has finite moments of all orders.
Using the reversibility of whole-plane SLEκ [40] [21] and the domain Markov property,
we then establish a similar upper bound for the probability that a whole-plane SLEκ passes
near any finite collection of points using the estimate for radial SLEκ . Using this estimate
and a proof similar to the radial case, we then prove that the lower Minkowski content of a
whole-plane SLEκ path in any compact set has finite moments of any order.
In Chapter 3, we establish a theory of Green’s functions for the backward SLEκ welding
for κ ≤ 4. Given any two points x, y > 0 we establish estimates for the probability of
the event that x and y are almost welded together in a way analogous to the estimate of
the probability that an SLEκ path passes near a point. Normalizing this estimate yields a
function G(x, y) which we call a backward SLEκ Green’s function. Moreover, we observed
that the process
Mt (x, y) = G(ft (x) −

√

√
κBt , κBt − ft (−y))

is a local martingale, and weighting the original probability measure by Mt (x, y) gives a
13

measure Px,y under which φ(x) = y almost surely.
Following the strategy in [37], we generalize this construction to get a family of functions
Ga,b (x, y) which serve as generalized Green’s functions for the BSLEκ welding. For any a, b,
we find powers p, q so that

a,b

Mt (x, y) = ft (x)p ft (−y)q Ga,b (ft (x) −

is a local martingale. If a, b ≤ −

√
√
κBt , κBt − ft (−y))

4+κ
, using this local martingale with the Girsanov theorem
2

a,b

yields a measure Px,y under which φ(x) = y almost surely.
For a = b = −4, we show that this measure gives insight into the BSLEκ welding in the
capacity parametrization in a way similar to what has been proven for forward chordal SLEκ .
Given any U ⊂ [0, ∞)×[0, ∞), we average the law of the welding conditioned to pass through
−4,−4

U by integrating Px,y

against G−4,−4 (x, y) over U . This yields a decomposition theorem

similar to those proven in [37] for the welding, and gives a measure PU which is absolutely
continuous with respect to the original probability measure. The resulting Radon-Nikodym
derivative is then shown to be the capacity time spent by the graph of the welding φ in U .

1.4

Generalizations of the Loewner equation

When Charles Loewner first introduced his differential equation in 1923 [18], it was a tool
used to study the Bieberbach conjecture. His version of the equation was the radial Loewner
equation, and the setting was entirely deterministic. The strategy was to solve (1.2) using
real valued continuous functions to study slit-domains as a way to answer extremal questions
about conformal maps in the unit disc. In [29], Oded Schramm recontextualized Loewner’s

14

ideas in order to study scaling limits of discrete random processes by using a Brownian motion
as the driving function. Along with introducing randomness, Schramm also introduced the
chordal Loewner ODE to study growth from a boundary point to another boundary point.
In the text [7], Lawler introduces generalized versions of both the chordal and radial
Loewner equation. Rather than being driven by a continuous function λt , the equations he
introduces are driven by a family of locally bounded Borel measures {µt }t≥0 on R (or ∂D
in the radial case) so that t → µt is weakly continuous. In the chordal case, the Loewner
equation driven by {µt }t≥0 is

∂t gt (z) =

1
dµt (u),
R gt (z) − u

g0 (z) = z.

Then for each t > 0, gt : H\Kt → H is conformal for some increasing family of H-hulls
(Kt )t≥0 with hcap(Kt ) = 0t µs (R)ds. This family of measures can be reparametrized so
that the corresponding family of hulls Kt satisfy hcap(Kt ) = 2t. For a real valued continuous
function λ, the regular chordal Loewner equation driven by λ is the above equation with
µt = δλt , where δx is the point mass measure at x.
In [19], Miller and Sheffield introduce a further generalization in the radial case which
they called measure driven Loewner evolution. They establish existence and uniqueness in
the radial setting, and then prove a one-to-one correspondence between families of D-hulls
and solutions to the measure driven Loewner evolution. Moreover, convergence on hulls was
shown to be equivalent to convergence of the corresponding measures.
In Chapter 4, we introduce measure driven chordal Loewner evolution. For a measure µ
on [0, ∞) × R with the appropriate assumptions, we show existence and uniqueness of the

15

solution to the integral equation

gt (z) − z =

2
dµ(s, u)
[0,t]×R gs (z) − u

for all z ∈ H. For a family of measures {µt }t≥0 as in Lawler’s definition, we define a measure
µ by dµ(t, u) = dµt (u)dt so that µ and {µt }t≥0 generate the same process..
If N is the family of measures which generate a measure driven chordal Loewner equation,
any µ ∈ N generates a family of H-hulls (Kt )t≥0 with hcap(Kt ) = 2t so that gt = gKt .
Conversely, we show that any family of H-hulls (Kt )t≥0 parametrized by capacity is the
resulting family of hulls for some measure µ ∈ N . Moreover, we endow N with a topology
and show that a family of measures µn → µ in N if and only if the domains H\Ktn → H\Kt
in the Carath´eodory topology (to be defined in Section 4.2) for each t, which is equivalent
to the conformal maps (gtn )−1 converging locally uniformly to gt−1 .

16

Chapter 2
Multipoint estimates for radial and
whole-plane SLE

2.1

Statement of results

The main theorem of this chapter is the multipoint estimate for radial SLEκ .
Theorem 1. Fix κ ∈ (0, 8). Let γ be a radial SLEκ in D from 1 to 0, let zk ∈ D\{0} for
k = 1, . . . , n, and let z0 = 1. Let yk = 1 − |zk | be the distance of each point to the boundary,
and define lk = min{|zk |, |zk − 1|, |zk − z1 |, . . . , |zk − zk−1 |}. Then there exists an absolute
constant Cn < ∞, depending on κ and n, so that
n

Pyk (rk ∧ lk )

P[∩n
k=1 {dist(γ, zk ) < rk }] ≤ Cn
k=1

Py (l )
k k

.

(2.1)

The function Py (x) used in the upper bound in Theorem 1 is defined by

Py (x) =





y α−(2−d) x2−d , x ≤ y



x α ,

,

x≥y

where d = 1 + κ/8 is the Hausdorff dimension of the path, and α = 8/κ − 1 is related to the
boundary exponent for SLEκ . This upper bound mixes the estimates for interior points and
17

points near the boundary. Roughly speaking, if the point zk is far from the boundary, the
term on the right hand side of (2.1) corresponding to zk will be (rk /lk )2−d . If zk is near the
boundary of the unit disc, then the corresponding term on the right side of (2.1) is (rk /lk )α .
If zk is close, but not too close, then the corresponding estimate is a mixture of the two.
The following Lemma about the functions Py is Lemma 2.1 in [24], and can be proven
with a case by case argument.
Lemma 1. For 0 ≤ x1 < x2 , 0 ≤ y1 ≤ y2 , 0 < x, and 0 ≤ y, we have
Py1 (x1 )
Py (x1 )
≤ 2
;
Py1 (x2 )
Py2 (x2 )

x1 α Py (x1 )
≤
≤
x2
Py (x2 )

x1 d−2 Px2 (x1 )
;
=
x2
Px2 (x2 )

y1 α−(2−d) Py1 (x)
≤
≤ 1.
y2
Py2 (x)
Theorem 1 can then be used to prove a similar estimate for whole-plane SLEκ :
Theorem 2. Fix κ ∈ (0, 8), and let γ ∗ be a whole-plane SLEκ trace from 0 to ∞. Let
z1 , . . . , zn ∈ C\{0}. For each k = 1, . . . , n, let 0 < rk < |zk | and define lk = min{|zk |, |zk −
z1 |, . . . , |zk − zk−1 |}. Then
n
∗
P[∩n
k=1 {dist(γ , zk ) < rk }] ≤ Cn
k=1

rk ∧ lk 2−d
.
lk

(2.2)

Note that the expression of this bound is simpler than Theorem 1, since there are no
boundary points with which to be concerned. Both of these estimates are used to prove the
18

following theorem about Minkowski content:
Theorem 3. Fix κ ∈ (0, 8).
a) Let γ be a radial SLEκ trace in D from 1 to 0. Then E[Contd (γ)n ] < ∞ for all n ∈ N.
b) Let γ ∗ be a whole-plane SLEκ trace from 0 to ∞, and suppose D ⊂ C is compact. Then
E[Contd (γ ∗ ∩ D)n ] < ∞ for every n ∈ N.
The chapter will be organized as follows. First we review preliminary information, including information about the covering space H∗ . We also review some information about
crosscuts, prime ends, and extremal length which will be used. Next, we provide one-point
estimates for radial SLE in the forms which will be useful for us. We then use these one-point
estimates to prove some key lemmas, followed by the proofs of the main theorems.

2.2
2.2.1

Preliminaries
Crosscuts and prime ends

In later sections, we will be studying the behavior of the radial SLE curve as it crosses many
interior curves, creating different components of the initial domain D. We need to introduce
some notation which will make it easier to distinguish which component is discussed at any
point in time. This is the same framework introduced in [23].
Recall that a crosscut in a simply connected domain D ⊂ C is a simple curve ρ : (a, b) →
D such that limt→a+ ρ(t) := ρ(a+ ) and limt→b− ρ(t) := ρ(b− ) both exist and are elements
of the boundary of D. Then ρ lies inside D, but the endpoints do not. The endpoints ρ(a+ )
and ρ(b− ) determine prime ends for the domain. Note that if f : D → D is a conformal

19

map, then f (ρ) is a crosscut in D . More information about crosscuts and prime ends can
be found in [1].
Note that if ρ is a crosscut in a domain D, then ρ divides D into two components.
Even more generally, let K ⊂ D be relatively closed. Let S be either a connected subset
of D\K or a prime end of D\K. We then define D(K; S) to be the component of D\K
which contains S. We also introduce the symbol D∗ (K, S) = D\ (K ∪ D(K; S)), which
is the union of the remaining components of D\K. This notation is useful for expressing
whether K seperates points. For example, if ρ ⊂ D is a crosscut which separates two points
z, w ∈ D, then D(ρ; z) = D(ρ; w). In fact, in this case, we have D(ρ; w) = D∗ (ρ; z), and
D(ρ; z) = D∗ (ρ; w).
Since we will be working with domains which have 0 as an interior point, and in particular
will be concerned with components containing 0, we will use D(K) and D∗ (K) to represent
D(K; 0) and D∗ (K; 0) respectively. Note that this is a departure from the notation in [23],
where the point being suppressed was the prime end ∞. Since we are working with radial SLE
rather than chordal SLE, the change is to reflect the fact that we are looking at components
with respect to the endoint of the SLE curve, which in this work will be the interior point
of 0.
For an example of how this notation will be used, let γ be a radial SLEκ curve in D from
1 to 0, and define Dt = D(γ[0, t]). This is the component of D\γ[0, t] which contains the
origin. If ρ is a crosscut in Dt , and z ∈ Dt \ρ, we will ask if z0 ∈ Dt (ρ). That is, does ρ
seperate z0 from 0 in Dt ? Suppose the circle ξ = {|z − z0 | = r} is contained in Dt \ρ. If ξ
doesn’t enclose 0, then D∗ (ξ) = {|z − z0 | < r}. Then D∗ (ξ) ⊂ Dt∗ (ρ) means the circle ξ is
seperated from 0 in Dt by ρ.
The next lemma is Lemma 2.1 in [23]:
20

˜ be simply connected domains in C. Let ρ either be a Jordan curve
Lemma 2. Let D ⊂ D
˜ which intersects ∂D or a crosscut in D.
˜ Let Z1 , Z2 be two connected subsets or prime
in D
˜ such that D(ρ;
˜ Zj ) is well defined for both j = 1, 2, and are nonequal. This means
ends of D
˜
˜
that D\ρ
is a neighborhood of both Z1 and Z2 in D, and Z1 is disconnected from Z2 in D
by ρ.
˜ Let Λ be the set of connected
Suppose that D is a neighborhood of Z1 and Z2 in D.
components of D ∩ ρ. Then there exists a unique λ1 ∈ Λ such that D(λ1 ; Z1 ) = D(λ1 ; Z2 ),
and if λ ∈ Λ such that D(λ; Z1 ) = D(λ; Z2 ), then D(λ1 ; Z1 ) ⊂ D(λ; Z1 ) and D(λ; Z2 ) ⊂
D(λ1 ; Z2 ).
The λ1 obtained in Lemma 2 will be referred to as the first subcrosscut of ρ to seperate
Z1 and Z2 in D. The conclusion of the lemma states that of all subcrosscuts of ρ in D which
disconnect Z1 and Z2 , λ1 is closest to Z1 in the sense that the component containing Z1 it
determines is contained in the component determined by any other such subcrosscut.

2.2.2

The covering space H∗ for D

The upper half-plane can be seen as a covering space for the unit disc under the exponential
map. Let H∗ be the cylinder defined by H∗ = {[z]∼ : z ∈ H}, where ∼ is the equivalence
relation z ∼ w if z − w ∈ 2πZ. Under this equivalence relation, the map ei : H∗ → D\{0}
defined by ei (z) = eiz is a conformal map. The boundary of H∗ is R∗ = R with the same
equivalence relation. We can treat 0 as the preimage of ∞ under ei , and ei0 = 1. The map
1
i

log : D\{0} → H∗ is also seen as a conformal map.
Any set K ⊂ H∗ can be viewed as a 2π-periodic set K + 2πZ ⊂ H. An H∗ -hull is a

set K ⊂ H∗ so that ei (K ) ⊂ D is a D-hull, and an H∗ -domain is a set H ⊂ H∗ so that

21

ei (H) ⊂ D is a D-domain. Therefore, an H∗ -domain is H∗ \K for an H∗ -domain K .
Radial SLEκ can be studied by looking at the covering space H∗ . Let λ ∈ C[0, ∞) and
z ∈ H∗ . The covering radial Loewner equation driven by λ is given by

∂t ht (z) = cot2 (ht (z) − λt ),

h0 (z) = z,

(2.3)

where cot2 (z) = cot(z/2). Note that if (gt ) is the radial Loewner evolution driven by λ, then
ht (z) = −i log gt (eiz ) satisfies (2.3). If λt =

√

κBt , where Bt is a standard one dimensional

Brownian motion, let γ be the preimage under ei of the radial SLEκ trace in D from 1 to 0
driven by λ. Then we call γ a radial SLEκ trace in H∗ from 0 to ∞, which is a 2π-periodic
family of paths in the cylinder H∗ . If H is an H∗ -domain with a prime end w0 , we can define
a radial SLEκ process in H from w0 to ∞ by mapping conformally from H to D = ei (H),
to D, then back to H∗ with 1i log.
To work in the cylinder H∗ , we must carefully define what distance and conformal radius
∗

mean in this domain. For z , w ∈ H which can be represented as z + 2πZ, w + 2πZ
∗

respectively for z, w ∈ H, then the distance from z to w in H is defined to be Euclidean
distance between the sets z + 2πZ and w + 2πZ in H. It will be written as |z − w |∗ to
distinguish from the distance between points in C. Then |z − w |∗ is the distance between
closest representatives of equivalence classes.
∗

Similarly, if A , B ⊂ H with A = A + 2πZ and B = B + 2πZ, then the distance from
A to B in H∗ is the Euclidean distance from A + 2πZ to B + 2πZ, and is denoted by
∗

distH∗ (A , B ). Given any z ∈ H with z = z + 2πZ, the ball of radius r centered at z is
denoted by B(z , r), and is represented in H by B(z, r) ∩ H + 2πZ. Note that for r < π, then
the representatives of B(z , r) are nonoverlapping in H.

22

In [2], several estimates are proven about one-point estimates with conformal radius
instead of distance, so we need to provide the definition of conformal radius in H∗ . Given
any simply connected domain D ⊂ C, and a point z ∈ D, the conformal radius of z in D
is defined by radD (z) = 1/|φ (z)|, where φ : D → D is conformal with φ(z) = 0. It follows
readily that radD (z) is a conformal invariant. That is, if φ : D → D is a conformal map and
z ∈ D, then radφ(D) (φ(z)) = |φ (z)|radD (z). Since φ : H∗ → D\{0} defined by φ(z ) = eiz
is a conformal map in the covering space, we can add ∞ into H∗ , thus adding 0 back into
the image of φ : H∗ ∪ {∞} → D so that φ is conformal and the cylinder is simply connected.
We can then use the conformal invarience formula to define conformal radius in the covering
space H∗ , and any other H∗ -domain.
If a set D is not connected, we define ΥD (z) to be the conformal radius of z in the
component of D in which z lies. This will be relevant when z becomes cut off from the
target of the SLE path by the curve. In this case, even though we define Dt to denote the
component of D\γ[0, t] containing the target by time t, we will still use the notation ΥDt (z).
For slightly easier calculations, we introduce the scaling ΥD (z) = radD (z)/2. From the
definition, a quick calculation verifies that radD (z) = 1 − |z|2 , and so ΥD (z) =

1 − |z|2
. For
2

any z ∈ H∗ , we can show that
ΥH∗ (z ) = sinh(y ),

(2.4)

where y = Im(z ). To see this, note that the conformal invarience definition shows that

radH∗ (z ) =

1
|ieiz

|

(1 − |eiz |2 ) = |e−iz | − |eiz | = ey − e−y ,

and dividing by 2 gives (2.4).
Remark 1: We will need to use applications of Koebe’s distortion theorem for the map
23

ei on sets in H∗ . For this to be valid, the set will have to sit inside B(z, r) + 2π for some
r < π so that ei can be restricted to a conformal map on a subdomain of C.
Remark 2: In [2], the covering space H∗ is defined slightly differently than we do by
assuming z ∼ w if z−w ∈ πZ. In this case, the associated conformal map is φ(z ) = e2iz , and
the same calculation as above gives radH∗ (z ) = 2 cosh(z ) sinh(z ) = sinh(2y ). However,
this slight change in coordinates only affects the estimates by a constant factor.

2.2.3

Extremal length and conformal transformations

Let dΩ (A, B) denote the extremal distance from A to B in Ω. For the definition of extremal
distance, see [1]. Note that this is distinct from the notations dist(A, B) and distH∗ (A, B),
both of which represent Euclidean distance. Define Λ(R) = dΩ ([−1, 0], [R, ∞)), where
R

ΩR = C\{[−1, 0] ∪ [R, ∞)}. By Teichm¨
uller’s theorem [1], this is maximal among doubly
connected domains in modulus (extrememal distance between boundary components) which
separate {−1, 0} and {w, ∞} with |w| = R. Moreover, Λ(R) ≤

1
log(16(R + 1)) for all
2π

values of R ≥ 1.
We are going to need to prove some estimates about extremal length which are conformally invariant, and so are more flexible than estimates involving Euclidean distance.
Moreover, we are going to look at extremal distance in the cylinder H∗ . Since H∗ looks
locally like H, if two sets are sufficiently close to each other, the extremal distance between
them should behave the same as the extremal distance in H.
Lemma 3. Let η be a crosscut in {z ∈ H : Re(z) > 0} with endpoints 0 < a < b. Define
r = sup{|z − a| : z ∈ η}. Then there is a constant C < ∞ so that, if r < a,

r
≤ Ce−πdH (η,(−∞,0]) .
a
24

(2.5)

Proof. Let Ω be the component of H\η whose boundary contains 0, and define Ω = Ω ∪
(0, a) ∪ (b, ∞). Let Ωdoub = Ω ∪ {z : z ∈ Ω}, and let η doub be the component of the
boundary of Ωdoub which does not contain 0, which is the closure of the double of η. Then
dH ((−∞, 0], η) = 2dΩdoub ((−∞, 0], η doub ).
Fix some w in the closure of η with |w − a| = r, and let ξ be the argument of w − a. Then
φ(z) =

(z − a)e−iξ /r − 1 is the affine transformation with φ(a) = −1, φ(w) = 0, and

ˆ = φ(Ωdoub ), then Ω
ˆ is a doubly connected domain seperating
|φ(0)| = |ae−iξ /r + 1|. If Ω
{−1, 0} from {φ(0), ∞}. By the conformal invarience of extremal distance and Teichm¨
uller’s
theorem, we have

doub ))) ≤
dΩdoub ((−∞, 0], η doub ) = dΩ
ˆ (φ((−∞, 0], φ(η

≤Λ

1
Λ |ae−iξ /r + 1|
2π

a
1
1
a
+1 ≤
ln(16(a/r + 1) + 1) ≤
ln 33 .
r
2π
2π
r

Rearranging this proves (2.5) for r small.

The following application of Koebe’s distortion theorem will be used repeatedly in the
next section to show that interior estimates are comparable after applying conformal maps.
We also derive a growth estimate for the inverse of the covering map for the cylinder.
Lemma 4.

a) Let D ⊂ C be a domain, and assume that B(z0 , R) ⊂ D. Suppose φ is a

conformal map defined on D, and let M =dist(z0 , ∂D). Suppose that r < R/7, and
define
˜=
R

R|φ (z0 )|
r|φ (z0 )|
, r˜ =
.
2
(1 + R/M )
(1 − r/M )2
25

Then
˜ ⊂ φ(B(z0 , R)),
φ(B(z0 , r)) ⊂ B(φ(z0 ), r˜) ⊂ B(φ(z0 ), R)
and there is an absolute constant C < ∞ so that

r
r˜
r
≤
≤C .
˜
R
R
R

b) Let D be a D-domain, and let H be an H∗ -domain with ei (H) = D. Let z0 ∈ D and
z0 ∈ H so that eiz0 = z0 . Then there is a constant C < ∞ so that if y0 = 1−|z0 | ≤ 1/2,
we have dist(z0 , ∂D) ≤ CdistH∗ (z0 , ∂H).
Proof. Applying the Growth Theorem to the univalent map f : D → C defined by

f (w) =

φ(M w + z0 ) − φ(z0 )
,
M φ (z0 )

we get that if |z − z0 | = ρ ∈ (0, M ), then
|φ(z) − φ(z0 )|
ρ/M
ρ/M
≤
≤
.
M |φ (z0 )|
(1 + ρ/M )2
(1 − ρ/M )2

(2.6)

˜ is M |φ (z0 )| times
Then r˜ is M |φ (z0 )| times the right hand side of (2.6) for ρ = r, and R
the left hand side for ρ = R. Then if
r (1 + R/M )2
r˜
=
˜
R (1 − r/M )2
R

26

is smaller than 1, we get the desired containments. If r < R/7 and R < M , then

r˜
1 4
≤
< 1.
˜
7 (6/7)2
R

Moreover,
1≤

4
(1 + R/M )2
≤
,
(1 − r/M )2
(6/7)2

which proves the comparability.
To prove part b) let, φ(z) = 1i log(z) be the map from D\{0} to H. Let r =dist(z0 , ∂D),
so that the assumption y0 ≤ 1/2 implies that a single valued branch of φ is conformal on
the open ball B(z0 , r). By Koebe’s 1/4 theorem,

B(z0 , r/4) ⊂ φ(B(z0 , r)) ⊂ H.

If w ∈ ∂H, this implies that |z0 − w |∗ ≥ r/4 = dist(z0 , ∂D)/4.

2.3

Interior and boundary estimates

The following estimate is Proposition 5.1 in [2].
Proposition 1. (Interior estimate 1) Let γ be a radial SLEκ curve in H∗ from 0 to ∞.
There exists constants 0 < c1 < c2 < ∞ so that if z ∈ H∗ with y = Im(z ) ≤ 1 and ≤ 1/2,
then
c1

y
|z |∗

α
2−d

≤ P Υ∞ (z ) ≤ Υ0 (z ) ≤ c2

where Υt (z ) is the conformal radius of z in H∗ \γ [0, t].
27

y
|z |∗

α
2−d ,

Lemma 5. (Interior estimate 2) Let γ be a radial SLEκ trace in H∗ from 0 to ∞. Then
there is a constant C < ∞ so that if z ∈ H∗ with y =Im(z ) ≤ ln(2) and r < y ,

P[distH∗ (z , γ ) ≤ r] ≤ C

y
|z |∗

α

r
y

2−d

=C

Py (r)
Py (|z |∗ )

.

Remark: By restricting to y ≤ ln(2), we are restricting slightly further than we are in
Proposition 1. We do this restriction so that we can use Lemma 4, part b). Later, we will
use part a) of Lemma 4 applied to a Loewner map to extend to the whole space.
Proof. Let z ∈ D so that eiz = z, denote the H∗ -hull determined by γ [0, t] by Ht , and let
Dt = ei (Ht ) for each t. By the definition of conformal radius of domains in H∗ , we have
ΥDt (z) = |(ei ) (z )|Υt (z ) = e−y Υt (z ) for all t < ∞. By Koebe’s 1/4 theorem and the
assumption y ≤ log(2), we have

Υt (z ) = ey ΥDt (z) ≤ Cdist(z, ∂Dt ).

Part b) of Lemma 4 implies that dist(z, ∂Dt ) ≤ CdistH∗ (z , ∂Ht ), and therefore

Υt (z ) ≤ CdistH∗ (z , ∂Ht ) ≤ CdistH∗ (z, γ [0, t]).

Taking t → ∞ gives Υ∞ (z ) ≤ cdistH∗ (z , γ ) for an absolute constant c < ∞.
By the above paragraph, for small , we have {distH∗ (z , γ ) < Υ0 (z ) } ⊂ {Υ∞ (z ) ≤
c
Υ0 (z )}. Therefore,

P[distH∗ (z , γ ) ≤ Υ0 (z ) ] ≤ C
c

28

y
|z |∗

α
2−d .

Renaming variables and picking

sufficiently small depending on z and r, we can let r =

Υ0 (z ) . Then
c

P[distH∗ (z , γ ) ≤ r] ≤ C

y
|z |∗

α

2−d
cr
≤C
Υ0 (z )

y
|z |∗

α

r
y

2−d

,

where the last inequality follows from (2.4), which says that Υ0 (z ) = ΥH∗ (z ) = sinh(y ) ≥
y . This is equal to Py (r)/Py (|z |∗ ), since |z |∗ ≥ y implies that Py (|z |∗ ) = |z |α
∗ and
r < y implies that Py (r) = (y )α−(2−d) r2−d .

The initial boundary estimate we will work with is Lemma 5.1 in [2].
Lemma 6. (Boundary estimate 0) There is a constant C < ∞ so that if γ is a radial SLEκ
curve in H∗ , x ∈ R∗ , and r < |x |∗ , then

P[dist(x , γ ) ≤ r] ≤ C

α
r
.
|x |∗

We want to modify this estimate into a more general and conformally invariant version
which can be applied in more general domains. In the next lemma, we will derive an estimate
involving the extremal distance between two crosscuts in the cylinder H∗ . This will be
determined by the domain between the crosscuts, and will be the same as the extremal
distance between a representation of each of them in H.
Lemma 7. Let D be any D domain, and let γ be radial SLEκ in D from a prime end w0 of
D to 0. Let ρ, η be crosscuts in D with small radius, η contained in the component of D\ρ

29

distinct from the component containing 0 and w0 . Then

P[γ ∩ η = ∅] ≤ Ce−απdD (ρ,η) .

(2.7)

Similarly, the inequality (2.7) holds when γ is radial SLEκ in an H∗ domain D from w0
to ∞, and ρ, η are non-self-intersecting 2π-periodic crosscuts in H∗ so that ρ separates w0
and ∞ from η.
Proof. In either case of the lemma, let φ be a conformal map from the domain to H∗ sending
w0 to 0. Then ρ = φ(ρ) is a non-self-intersecting crosscut in H∗ which separates 0, ∞
from η = φ(η), and φ(γ) = γ˜ is a time changed radial SLEκ curve in H∗ . Let η0 be the
representative for η in H with η0 ⊂ {z ∈ H : 0 < Re(z) ≤ 2π}, and define ρ0 similarly.
Denote the endpoints of η0 by 0 < a < b < 2π, and let x = min{a, 2π − b}. If x = a,
define r = sup{|z − a| : z ∈ η0 }. If x = 2π − b, let r = sup{|z − b| : z ∈ η0 }. Either way,
η0 ⊂ B(x, r), and so by Lemma 6, we have

P[γ ∩ η = ∅] = P[˜
γ ∩ η = ∅] ≤ P[dist(˜
γ , x) ≤ r] ≤ C

r α
.
x

Applying Lemma 3 to η0 , we have
r
≤ C e−πdH (η0 ,(−∞,0]) ∨ e−πdH (η0 ,[2π,∞)) ≤ Ce−πdH (ρ0 ,η0 ) ,
x

where the last inequality is due to the comparison principle of extremal length. Combining
these two inequalities gives

P[γ ∩ η = ∅] ≤ Ce−απdH (ρ0 ,η0 ) = Ce−απdD (ρ,η) ,
30

where the last equality is from the conformal invariance of extremal length, and since it is
the domain between the crosscuts which determines the length.

Now we can combine the interior and boundary estimates into the general one-point
estimate for the cylinder. We now also remove the assumption that y0 ≤ ln(2) to extend the
estimate to the whole space.
Lemma 8 (One-point estimate for cylinder). Suppose that H is an H∗ domain, and γ is
radial SLEκ in H from a prime end w0 to ∞. Fix z0 ∈ H∗ with imaginary part y0 , and let
π > R > r > 0. Define ρ = {z ∈ H∗ : |z − z0 |∗ = R}, η = {z ∈ H∗ : |z − z0 |∗ = r}.
Moreover, suppose that {z ∈ H∗ : |z − z0 |∗ ≤ R} ⊂ D and that w0 ∈
/ {x ∈ R∗ : |z0 − x |∗ <
R}. Then
P[γ ∩ η = ∅] ≤ C

Py (r)
0

Py (R)

.

0

Proof. Observe that, by assumption, the representatives for ρ in H under the equivalence
relation consists of multiple disconnected components, rather than having overlap. This
means any of the covering maps are locally conformal at each piece of ρ, and everything
inside of ρ, so Lemma 4 will be applicable.
The proof breaks down into three cases, each depending on how far from R∗ is the point
z0 . The first case is the far away case, when y0 > R. Also, we must first assume that
y0 ≤ ln(2). Let hH : H → H∗ be the canonical conformal map taking w0 to 0 and ∞ to ∞,
so that γ˜ = hH (γ ) is a time-changed radial SLEκ curve from 0 to ∞ . Let z˜0 = hH (z0 ),
and let y˜0 = Im(˜
z0 ). Since hH is conformal on a simply connected neighborhood of any
component of ρ, Lemma 4 a) implies that, assuming r < R/7 without loss of generality (else

31

we modify the constant by a factor of 7),

z0 , γ˜ ) ≤ r˜]
P[γ ∩ η = ∅] = P[˜
γ ∩ hH (η) = ∅] ≤ P[distH∗ (˜
Py˜0 (˜
r)

Py˜ (˜
r)
≤C
≤C 0
=C
˜
Py˜0 (|˜
z0 |∗ )
Py˜0 (R)

r˜ 2−d
r 2−d
≤C
,
˜
R
R

˜ are defined as in Lemma 4 with respect to the conformal map r, R, and hH .
where r˜, R
Assume now that z0 ∈ H is arbitrary, and let ht : H\γ [0, t] → H be the Loewner
maps for the radial SLEκ process in H. For any T > 0, define γ T (t) = γ (T + t), so that
conditioned on FT , γ T is a radial SLEκ trace from γ (T ) to ∞. Define zt = ht (z0 ), and
define a stopping time by τ = inf{t ≥ 0 : Im(zt ) ≤ ln(2)}. Define rτ , Rτ as in Lemma 4 at
z0 with respect to r, R, hτ . Then Lemma 4 implies that

P[distH∗ (zτ

, γτ )

< r|Fτ , τ < ∞] ≤ C

rτ 2−d
r 2−d
≤C
.
Rτ
R

Since P[τ < ∞] ≤ 1, this proves the lemma for all z0 ∈ H.
For case 2, we consider the close case, 0 < y0 < r. We will use the boundary estimates
r α
to derive an upper bound of
. By modifying the constant slightly, we can assume that
R
R > 4r. Then in order to cross from ρ to η, γ must also pass through ρ = {z ∈ H∗ :
|z − Re(z0 )|∗ = R/2} and η = {z ∈ H∗ : |z − Re(z0 )|∗ = 2r}, which are two semicircles
such that dD (ρ , η ) = (1/π) log(R/4r). Using Lemma 7, we get that

P[γ ∩ η = ∅] ≤ P[γ ∩ η = ∅] ≤ Ce−απdD (η ,ρ ) = C

r α
.
R

Lastly, we have to mix interior and boundary estimates in the middle distance case when

32

0 < r < y0 < R. Let ρ = {z ∈ H∗ : |z − z0 |∗ = y0 } which is a circle tangent to
R∗ between η and ρ. Let T = inf{t > 0 : γ (t) ∈ ρ }, which is a stopping time so that
{γ ∩ η = ∅} ⊂ {T < ∞}. Using case 2, we can see that

P[T < ∞] ≤ C

Py (y0 )
0
Py (R)

.

0

Define γ T (t) = γ (T + t). By the domain Markov property and the first case, we can see
that
P[γ T

∩ η = ∅|γ [0, T ], T < ∞] ≤ C

Py (r)
0

Py (y0 )
0

.

Combining these two inequalites gives case 3.

Using the Growth Theorem estimate again, we can prove the analogous estimate for
radial SLE in the disc.
Lemma 9 (One-point estimate for disc). Let D be a D domain with a prime end w0 , and
let γ be a radial SLEκ curve in D from w0 to 0. Let z0 ∈ D, y0 = 1 − |z0 |. Suppose that 0 <
r < R < dist(z0 , {0, w0 }). Let ρ = {z ∈ D : |z − z0 | = R}, and η = {z ∈ D : |z − z0 | = r},
and assume that {z ∈ D : |z − z0 | ≤ R} ⊂ D. Then

P[γ ∩ η = ∅] ≤ C

Py0 (r)
.
Py0 (R)

Proof. Let φ : D\{0} → H∗ be defined by φ(z) = 1i log(z), and suppose z0 ∈ H := φ(D)
so that eiz0 = z0 . The proof will break into the same 3 cases as in the proof of Lemma 8:

33

R < y0 , r > y0 , and r < y0 < R. In the first two cases, we want to prove that

P[dist(z0 , γ)] ≤ C

r p
R

for p ∈ {α, 2 − d}, so it will suffice to prove an upper bound for r/R which will work for both
cases simultaneously. The third case can be handed similarly to the third case in Lemma 8.
By assumption, φ is a conformal map on a domain which has positive distance from
˜ as in Lemma 4
B(z0 , R). Let γ be a radial SLEκ trace in H from φ(w0 ) to ∞. Define r˜, R
with respect to φ and z0 . If r < R/7, the inclusions in Lemma 4, part a), imply that in the
˜ and in the boundary
interior case, we can apply the interior estimate of Lemma 8 to z0 , r˜, R,
case we can apply the boundary estimate. In either case,

P[dist(z0 , γ) ≤ r] ≤ P[distH∗ (z0 , γ ) ≤ r˜] ≤ C

r˜ p
r p
,
≤C
˜
R
R

where p ∈ {α, 2 − d} is chosen appropriately.

Lemma 10. Let D be a D domain with a prime end w0 , and let γ be a radial SLEκ curve
in D from w0 to 0. Let ρ be a crosscut in D such that D∗ (ρ) is not a neighborhood of w0
˜ be a domain that contains D, and ρ˜ be a subset of D
˜ that
in D, and let S ⊂ D∗ (ρ). Let D
˜ which intersects ∂D or a crosscut in D.
˜
contains ρ. Let η˜ be either a Jordan curve in D
˜ Then
Suppose that η˜ disconnects ρ˜ from S in D.

−απd ˜ (˜
ρ,˜
η)
D
.

P[γ ∩ S = ∅] ≤ Ce

Proof. By Lemma 2, η˜ has a subcrosscut η in D which disconnects S from ρ. Since S ⊂
34

D∗ (ρ), we have η ⊂ D∗ (ρ) and S ⊂ D∗ (η). Therefore, D(ρ; η) = D∗ (ρ) is neither a
neighborhood of 0 nor w0 in D. Using the boundary estimate from Lemma 7, we see that

P[γ ∩ S = ∅] ≤ P[γ ∩ D∗ (η) = ∅] ≤ Ce−απdD (ρ,η) ≤ C

−απd ˜ (˜
ρ,˜
η)
D
.

The last inequality follows from the comparison principle for extremal length, since any path
˜ which connects ρ˜ and η˜.
in D connecting ρ and η is a subarc of itself contained in D

2.4

Components of crosscuts

Before we state the main theorem of this section, we will introduce the notation to be used.
Let Ft be the right continuous filtration determined by the radial SLE curve γ. For any set
S ⊂ D, let τS = inf{t ≥ 0 : γ(t) ∈ S}. For any stopping time τ , define γ τ (t) = γ(τ + t).
Define Dt = D(γ[0, t]).
Theorem 4. Let γ be a radial SLEκ curve in D from 1 to 0. Suppose that z0 , z1 , . . . , zm ∈
D\{0, 1}. For each zj , let 0 < rj ≤ Rj , and define the circles ξˆj = {|z − zj | = Rj } and
ξj = {|z − zj | = rj }. Assume that neither 0 nor 1 are contained in D∗ (ξˆj ; zj ) for each j,
and that D∗ (ξˆj ) ∩ D∗ (ξˆi ) = ∅ for j = j. Let r0 ∈ (0, r0 ) and define ξ0 = {|z − z0 | = r0 }.
Define the event

E = {τξ0 < τξˆ ≤ τξ1 < · · · < τξˆ ≤ τξm < τξ < ∞}.
m
1
0

35

If yj = 1 − |zj |, then
P[E|Fτξ ] ≤ C m
0

m P (r )
yj j
r0 α/4
.
R0
Pyj (Rj )
j=1

Note that the assumptions here imply that neither the start point, 1, nor the endpoint
0, of the SLE curve are enclosed by the discs ξˆj . We also assume interiors of circles with
different centers have no overlap, and that the boundary circles do not meet. The proof is
similar to the proof of Theorem 3.1 in [23], but for completeness we include complete details.
Proof. Consider the discs ξ which intersect the boundary. We know that the probability
that γ hits the points in ξ ∩ ∂D is equal to 0, and so τξ = τξ∩D a.s. Therefore, we can assume
that each ξ is either a Jordan curve or a crosscut in D. For each j = 0, 1, . . . , m, let τj = τξj
and τˆj = τξˆ , and define τm+1 = τξ .
j
0
By the Domain Markov Property of SLE and Lemma 9, we see that

P[τj < ∞|Fτˆj ] ≤ C

Pyj (rj )
Pyj (Rj )

.

(2.8)

Combining these together gives
m

P[E|Fτ0 ] ≤

Cm
j=1

Pyj (rj )
Pyj (Rj )

.

If r0 = R0 , then we are done. Suppose that R0 > r0 . Create a new arc ρ = {z ∈ D :
|z − z0 | =

√

R0 r0 }, so that ρ is either a Jordan curve or a crosscut in D between ξ0 and ξˆ0 .

Therefore, we know that

log(R0 /r0 )
dD (ρ, ξ0 ), dD (ρ, ξˆ0 ) ≥
.
4π

36

(2.9)

Note that ρ seperates ξ0 from 0. In the following argument, we will need to keep track of
how the D-domains Dt are divided by ρ at any particular time. Let T = inf{t ≥ 0 : ξ0 ⊂ Dt }.
Then if τ0 ≤ t < T , ξ0 is a connected subset of Dt . In this case, since the starting point 1 is
outside of ξˆ0 and γ intesercts ξ0 , it must be that γ intersects ρ, and so ρ intersects ∂Dt . By
Lemma 2, there is a first subcrosscut of ρ in Dt , to be denoted ρt , which seperates ξ0 from
0 for each τ0 ≤ t < T .
Now, we need to break the event E into several cases based on the behavior of the curve
γ as it intersects the circles in the correct order. Let I = {(j, j + 1) : 0 ≤ j ≤ m} ∪ {(j, j) :
1 ≤ j ≤ m}, and define a sequence of events {Ai : i ∈ I} by
1. A(0,1) = {T > τ0 } ∩ {D∗ (ξ1 ) ⊂ Dτ∗0 (ρτ0 )}
2. A(j,j) = {T > τj } ∩ {D∗ (ξj ) ⊂ Dτj−1 (ρτj−1 )} ∩ {D∗ (ξj ) ⊂ Dτ∗ (ρτj )}, 1 ≤ j ≤ m.
j
3. A(j,j+1) = {T > τj } ∩ {D∗ (ξj ) ⊂ Dτj (ρτj )} ∩ {D∗ (ξj+1 ) ⊂ Dτ∗ (ρτj )}, 1 ≤ j ≤ m − 1.
j
4. A(m,m+1) = {T > τm } ∩ {D∗ (ξm ) ⊂ Dτm (ρτm )}.
Observe that for each j, the events A(j,j) , A(j,j+1) are Fτj measurable. What I claim is that

E ⊂ ∪i∈I Ai .

(2.10)

To see this, observe that A(m,m+1) is the event that at time τm , ξm lies outside of ρ, relative
to 0 and the path γ. If that does not happen, then ξm must lie inside ρ at time τm . Now
suppose that A(m,m) does not happen, which is the event that at time τm−1 , ξm lies outside
of ρ, but at time τm , ξm lies inside of ρ. Then it must be that ξm lies inside of ρ at
τm−1 . Proceeding along this way inductively proves (2.10). Complete details can be found

37

in Section 2.7. Now it will suffice to show that

P[E ∩ Ai |Fτ0 ] ≤

Cm

m P (r )
yj j
r0 α/4
R0
Pyj (Rj )

(2.11)

j=1

for each i ∈ I. We will break it down to the four cases i = (m, m + 1), (j, j), (j, j + 1), and
(0, 1). In all of these cases, we will use the convention γ T (t) = γ(T + t) for any T, t ≥ 0.
Case (0, 1). Suppose that A(0,1) occurs, and that τ0 < τˆ1 . First, we claim that ξˆ1 ⊂
D∗τ0 (ρτ0 ). Note that both ξˆ1 and D∗ (ξ1 ) are contained in D∗ (ξˆ1 ) ∪ ξˆ1 , which is a connected
subset of the disjoint union Dτ0 (ρτ0 ) ∪ Dτ∗0 (ρτ0 ). By assuming that the event A(0,1) occurs,
however, we can conclude that they must both be contained in Dτ∗0 (ρτ0 ), which proves the
claim. Also, note that ρ disconnects ξˆ1 from ξ0 in D, and ρ must intersect ∂Dτ0 . By Lemma
2, there is a subcrosscut ρτ0 of ρ which is first to seperate ξˆ1 from ξ0 in the domain Dτ0 .
Since both ξˆ1 and ξ0 lie in Dτ∗0 (ρτ0 ), so does ρτ0 . Note that this implies that ρτ0 = ρτ0 , and
that Dτ∗0 (ρτ0 ) ⊂ Dτ∗0 (ρτ0 ). Since ρτ0 was defined to be the first subcrosscut of ρ in Dτ0 that
disconnects ξ0 from 0, and since Dτ∗0 (ρτ0 ) is contained in the domain determined by ρτ0 , it
cannot be that ρτ0 disconnects ξ0 from 0. Therefore, we conclude that ξ0 ⊂ Dτ0 (ρτ0 ) and
ξˆ1 ⊂ Dτ∗0 (ρτ0 ).
Observe that D∗ (ξ0 ) is a connected subset of Dτ0 \ρτ0 , and contains ξ0 and a curve which
approaches γ(τ0 ) ∈ ξ0 . Therefore,

Dτ0 (ρτ0 ; γ(τ0 )) = Dτ0 (ρτ0 ; ξ0 ) = Dτ0 (ρτ0 ).

It follows that Dτ0 (ρτ0 ; ξˆ1 ) = Dτ∗0 (ρτ0 ) is not a neighborhood of γ(τ0 ) = γ τ0 (0), where γ τ0
(conditioned on Fτ0 ) is a radial SLEκ curve in the D-domain Dτ0 . Since τ0 < τˆ1 , the event

38

{ˆ
τ1 < ∞} implies that the conditioned SLE curve γ τ0 visits ξˆ1 . Since ξˆ0 disconnects ξˆ1 from
ρτ0 ⊂ ρ in D, and ξˆ0 intersects ∂Dτ0 , we can apply Lemma 10 to conclude that
ˆ

P[ˆ
τ1 < ∞|Fτ0 , A(0,1) , τ0 < τˆ1 ] ≤ Ce−αdD (ρ,ξ0 ) ≤ C

r0 α/4
.
R0

Note that the second inequality follows from (2.9). Combining the above inequality with
inequality (2.4) proves that inequality (2.11) holds for the case i = (0, 1).
Case (j, j + 1), for 1 ≤ j ≤ m − 1. Suppose that A(j,j+1) occurs, and τj < τˆj+1 . By the
same argument used in the case for A(0,1) , we can conclude that there exists a subcrosscut of
ρ, which we will call ρτ , which disconnects ξˆj+1 from ξ0 in Dτj . It follows that Dτ∗ (ρτ ) ⊂
j
j
j
Dτ∗ (ρτj ), and then we can conclude that ξ0 ⊂ Dτj (ρτ ) and ξˆj+1 ⊂ Dτ∗ (ρτ ). Since D∗ (ξj )
j
j
j
j
is a connected subset of Dτj (ρτj ) and contains a curve approaching γ(τj ) ∈ ξj , we can see
that Dτj (ρτj ; γ(τj )) = Dτj (ρτj ; D∗ (ξj )) = Dτj (ρτj ). Therefore, Dτ∗ (ρτ ) ⊂ Dτ∗ (ρτj ) is not
j
j
j
τ
a neighborhood of γ(τj ) = γ j (0).
τ
Since τj < τˆj+1 , the event {ˆ
τj+1 < ∞} implies that γ j , the radial SLEκ curve in Dτj

after conditioning on Fτj , visits ξˆj+1 . Since ξˆ0 disconnects ξˆj+1 from ρ, and therefore from
ρτ , in D, Lemma 10 implies that
j

ˆ

P[ˆ
τj+1 < ∞|Fτj , A(j,j+1) , τj < τˆj+1 ] ≤ Ce−απdD (ρ,ξ0 ) ≤ C

r0 α/4
.
R0

As in the above case, this proves inequality (2.11) for the case i = (j, j + 1) when 1 ≤ j ≤
m − 1.
Case (m, m + 1). Suppose that τm < τm+1 , and that A(m,m+1) occurs. The set D∗ (ξm ) is
connected and contained in Dτm \ρτm , and also γ(τm ) ∈ ξm . This implies that Dτm (ρτm ; γ(τm )) =

39

Dτm (ρτm ; D∗ (ξm )) = Dτm (ρτm ). Therefore, Dτ∗m (ρτm ) is not a neighborhood of γ τm (0) =
γ(τm ) in Dτm . Since we are assuming that τm < τm+1 , the event {τm+1 < ∞} implies that
the curve γ τm , which after conditioning on Fτm is a radial SLEκ in Dτm from γ(τm ) to 0,
must visit ξ0 ⊂ Dτ∗m (ρτm ). Since ξ0 disconnects ξ0 from ρ in D and intersects ∂Dτm , we
can apply Lemma 10 to show that

P[τm+1 < ∞|Fτm , A(m,m+1) , τm < τm+1 ] ≤ Ce−απdD (ξ0 ,ρ) ≤ C

r0 α/4
.
R0

Case (j, j), for 1 ≤ j ≤ m. We now prove inequality (2.11) for A(j,j) . Fix a j in
{1, . . . , m}, and define σj = inf{t ≥ τj−1 : D∗ (ξj ) ⊂ Dt∗ (ρt )}. This can be seen as the first
time after τj−1 that the SLE curve γ hits the crosscut ρ on the “correct” side of ξˆj , i.e. γ cuts
the disc off from 0. There are a few observations to make about σj which follow from Lemma
11 (in Section 2.7). First, σj is an Ft -stopping time. If σj < ∞, then D∗ (ξj ) ⊂ Dσ∗ (ρσj ).
j
If the event A(j,j) occurs, then so does {τj−1 < σj < τj }. Finally, if τj−1 < σj < ∞,
then it must be that γ(σj ) is an endpoint of ρσj . This implies that Dσ∗ (ρσj ) is neither a
j
neighborhood of γ(σj ) nor of 0.
We define two events which seperate the event A(j,j) . Define

F< = {σj < τˆj }, and F≥ = {τj > σj ≥ τˆj }.

Notice that A(j,j) ⊂ F< ∪ F≥ , so if we can prove (2.11) for both F< and F≥ instead of A(j,j) ,
the case will be proven.
First, assume that F< happens. Then D∗ (ξˆj )∪ ξˆj is a connected subset of (D\γ[0, ρσj ])\ρ
that contains D∗ (ξj ), and so ξˆj ⊂ Dσ∗ (ρσj ; D∗ (ξj )) = Dσ∗ (ρσj ). Since ξˆ0 disconnects ρ from
j
j

40

ξˆj in D, Lemma 10 and (2.9) imply that

ˆ

P[ˆ
τj < ∞|Fσj , F< ] ≤ Ce−απdD (ρ,ξ0 ) ≤ C

r0 α/4
.
R0

This implies that
P[τj < ∞, F< |Fτj−1 ] ≤ C

r0
R)

α/4

Pyj (rj )
Pyj (Rj ))

.

Next, we assume that F≥ occurs, which is the more difficult case. Define N = log(Rj /rj ) ,
where · is the integer valued ceiling function. In the event F≥ , the SLE curve passes through
the outer circle ξˆj , then hits the crosscut ρ before returning to hit the inner circle ξj . What
we need to do is divide the annulus {rj ≤ |z − zj | < Rj } into N subannuli until we can
identify the last subcircle to be crossed before the time σj .
j 1/N

Define the circles ζk = {|z − zj | = RjN −k rj

} for 0 ≤ k ≤ N . Note that ζ0 = ξˆj ,

ζN = ξj , and a higher index k indicates that ζk is more deeply nested inside ξˆj . Then
F≥ ⊂ ∪N
k=1 Fk , where
Fk = {τζ

k−1

≤ σj < τξ }.
ζ
k

If the event Fk occurs, then ζk ⊂ Dσ∗ (ρσj ). This is because D∗ (ζk )∪ζk is a connected subset
j
of D\γ[0, σj ] \ρ which contains ζk , D∗ (ξj ), and Dσ∗ (ρσj ). By Lemma 10 and inequality
j
(2.9), we conclude that

ˆ

P[τζ < ∞|Fσj , Fk ] ≤ Ce−απdD (ρ,ζk−1 ) ≤ Ce−απ(dD (ρ,ξ0 )+dD (ζ0 ,ζk−1 ))
k

≤C

r0 α/4
R0

rj α(k−1))/2N
.
Rj

41

(2.12)

By Lemma 9, we get that

P[Fk |Fτj−1 , τj−1 < τˆj ] ≤ C

P[τj < ∞|Fζ , Fk ] ≤ C
k

Pyj (RjN −k+1 rk−1 )1/N
Pyj (Rj )
Pyj (rj )
Pyj (RjN −k rjk )1/N

,

.

Combining the above three inequalities and the upper bound in Lemma 1 gives

P[τj < ∞, Fk |Fτj−1 , τj−1 < τˆj ] ≤ C

r0 α/4
R0

α(k−1)
2N

rj
Rj

rj −α/N Pyj (rj )
.
Rj
Pyj (Rj )

Since F≥ ⊂ ∪N
k=1 Fk , adding up the above inequality yields

P[τj < ∞, F≥ |Fτj−1 , τj−1 < τˆj ] ≤ C

r0 α/4 Pyj (rj )
R0
Pyj (Rj )

rj −α/N 1 − (rj /Rj )α/2
.
Rj
1 − (rj /Rj )α/2N

In Section 2.7, we prove that
rj −α/N 1 − (rj /Rj )α/2
eα
≤
,
Rj
1 − (rj /Rj )α/2N
1 − e−α/4

(2.13)

and so we get

P[τj < ∞, F≥ |Fτj−1 , τj−1 < τˆj ] ≤ C

r0 α/4 Pyj (rj )
,
R0
Pyj (Rj )

which is the same bound as that acheived for F< , though with a different constant. Com-

42

bining these two inequalities gives

P[τj < ∞, A(j,j) |Fτj−1 , τj−1 < τˆj ] ≤ C

r0 −α/4 Pyj (rj )
,
R0
Pyj (Rj )

which completes the proof of (2.11) in the final case, and so the theorem follows.

2.5

Concentric circles

Let Ξ be a family of mutually disjoint circles in C with centers in D\{0}, none of which pass
through or enclose 0 or 1. We can define a partial order on Ξ by ξ1 < ξ2 if ξ2 is enclosed by
ξ1 . Note that the larger circle has a smaller radius than the larger circle because the order
is determined by visiting time. If ξ1 < ξ2 , any continuous path which hits both circles must
pass through ξ1 before it hits ξ2 . When γ is an SLE curve in D starting from 0, ξ1 < ξ2
means that τξ1 < τξ2 , assuming γ passes through ξ2 . Also, observe that circles in Ξ are not
necessarily contained in D. In fact, we want to account for circles in Ξ which have center in
the boundary as well.
Further, we assume that Ξ has a partition ∪e∈E Ξe with the following properties:
1. For each e ∈ E, the elements of Ξe are concentric circles whose radii form a geometric
sequence with a common ratio of 1/4. For each e, let the common center be ze . Note
that elements of Ξe are totally ordered. Let Re be the radius of the smallest circle (in
the ordering on Ξ), and let re be the radius of the largest circle. Then there is some
integer M ≥ 0 with Re = re 4M .
2. Let Ae = {z ∈ C : re ≤ |z − ze | ≤ Re } denote the closed annulus containing all of
the circles in Ξe . Then we assume that the collection of annuli {Ae }e∈E is mutually
43

disjoint.
We make a couple of quick remarks about the generality of this assumption. It may be that
|Ξe | = 1, in which case re = Re , and the annulus Ae is just the single circle contained in Ξe .
Also, if e1 = e2 , that does not necessarily mean that ze1 = ze2 . There can be multiple sets of
concentric circles in Ξ with the same center with gaps between them. In this case, if members
of Ξe2 have the smaller radii, we have Re2 < re1 . Also, there can be two components of the
partition Ξe1 , Ξe2 so that each element in Ξe2 is ordered larger than each element of Ξe1 .
In this case, the annulus Ae2 is contained in the bounded component of C\Ae1 .
Theorem 5. Let Ξ be a family of circles with the properties listed above, and assume that γ
is a radial SLEκ curve in D from 1 to 0. For each e ∈ E, let ye = 1 − |ze |. Then there exists
a constant C|E| < ∞, which only depends on κ and the size of the partition |E|, so that

P[∩ξ∈Ξ {γ ∩ ξ = ∅}] ≤ C|E|
e∈E

Pye (re )
.
Pye (Re )

The strategy is to consider all possible orders σ that the SLE curve γ can visit all elements
of Ξ. Under σ, γ may pass through several elements of a family Ξe0 , leave and visit other
Ξe ’s, before returning to pass through the more inner circles in Ξe1 . Theorem 4 provides an
estimate of the price paid by γ in order to return to the interior circles of Ξe0 . This gives
an estimate for the probability of ∩ξ {γ ∩ ξ = ∅} in the prescribed order σ. We then add up
over all appropriate orders σ and show that the constant only depends on |E|.
Proof. Define S to be the set of permutations σ : {1, 2, . . . , |Ξ|} → Ξ such that ξ1 < ξ2 implies
σ −1 (ξ1 ) < σ −1 (ξ2 ). Then S is the set of viable orders in which γ can visit the elements of Ξ
for the first time. For an ordering σ ∈ S, σ(j) ∈ Ξ is the j-th circle visited by γ. Define the

44

event E σ = {τσ(1) < · · · < τσ(|Ξ|) < ∞}. Then E := ∩ξ∈Ξ {γ ∩ ξ = ∅} = ∪σ∈S E σ . What
we need to do is bound P[E σ ].
Fix some σ ∈ S. Our first goal is to create a subpartition of {Ξe }e∈E into {Ξi }i∈I ,
where the elements of Ξi receive first visits from γ without interruption in the event E σ .
e . Define
For each e ∈ E, let Ne = |Ξe | − 1, and label the elements of Ξe by ξ0e < · · · < ξN
e

Je ⊂ {0, 1, . . . , Ne } by

e ) + 1} ∪ {0}.
Je = {1 ≤ n ≤ Ne : σ −1 (ξne ) > σ −1 (ξn−1

e , the curve visits other new circles
Then n is a nonzero element of Je if, after γ visits ξn−1

in Ξ before ξne . That is, there is some ξ ∈ ∪e =e Ξe such that τξ e
< τξ < τξ e . Order the
n
n−1
elements of Je by 0 = se (0) < se (1) < · · · < se (Me ), where Me = |Je | − 1 is the number of
times that the progress of γ through Ξe is interrupted. Define se (Me + 1) = Ne + 1. Using
this framework, each Ξe can be partitioned into Me + 1 subsets

Ξ(e,j) = {ξne : se (j) ≤ n ≤ se (j + 1) − 1}, 0 ≤ j ≤ Me .

These are the elements of Ξe which are visited without interruption. Let I = {(e, j) : e ∈
E, 0 ≤ j ≤ Me }. Then {Ξi }i∈I is a finer partition with the desired properties.
For i ∈ I, let i = (ei , j). We need to do some relableing. Let

e

zi = zei , yi = 1 − |zi |, min{Ξi } = ξs i (j) ,
ei
Pyi (Rmax{Ξ } )
e
i ,
max{Ξi } = ξs i (j+1)−1 , Pi =
ei
Pyi (Rmin{Ξ } )
i

45

where Rmax{Ξ } is the radius of max{Ξi }, and Rmin{Ξ } is the radius of min{Ξi }. By Lemma
i
i
9, we can see that
P[τmax{Ξ } < ∞|Fτmin{Ξ } ] ≤ CPi .
i
i

(2.14)

For e ∈ E, let Pe = Pye (re )/Pye (Re ). We claim that for each e ∈ E, we have
Me

P(e,j) ≤ 4αMe Pe .

(2.15)

j=0

This follows from Lemma 1, and the details are provided in Section 2.7.
Observe that |I| =

e∈E (Me

+ 1) is the number of uninterrupted sequences of circles

visited by γ under σ. Then σ induces a map σ
ˆ : {1, . . . , |I|} → I so that if n1 < n2 , we have
max{σ −1 (Ξσˆ (n ) )} < min{σ −1 (Ξσˆ (n ) )}, and n1 = n2 − 1 implies max{σ −1 (Ξσˆ (n ) )} =
1

2

1

min{σ −1 (Ξσˆ (n ) )} − 1. In other words, σ
ˆ is the order that the families Ξi are visited. In
2

particular, we can rewrite the event E σ as

E σ = {τmin Ξ

σ
ˆ (1)

< τmax Ξ

σ
ˆ (1)

< · · · < τmin Ξ

σ
ˆ (|I|)

< τmax Ξ

σ
ˆ (|I|)

< ∞}.

We will use Theorem 4 to estimate the probability of this event.
Fix some e0 ∈ E, and let nj = σ
ˆ −1 ((e0 , j)) for 0 ≤ j ≤ Me0 . In the event E σ , the family
Ξ(e ,j) is the nj -th family to be visited by the curve. For 0 ≤ j ≤ Me − 1, nj+1 ≥ nj + 2
0
since at least one other family not contained in Ξe0 must be hit by the curve between them.
Fix 0 ≤ j ≤ Me0 − 1, and let m = nj+1 − nj − 1 ≥ 1. We are going to apply Theorem 4 to
the family
• ξˆ0 = min Ξe0 , ξ0 = max Ξ(e ,j) = max Ξσˆ (n ) , and ξ0 = min Ξ(e ,j+1) = min Ξσˆ (n )
0
0
j
j+1

46

• ξˆk = min Ξσˆ (n +k) , and ξk = max Ξσˆ (n +k) for 1 ≤ k ≤ m.
j
j
In plain words, m is the number of other families {Ξi } first visited by γ between first visits
of the j-th level and the (j + 1)-st level of the family Ξe0 . The curves ξk , ξˆk are the k-th
family first visited before returning to Ξ(e ,j+1) .
0
σ
by
Define an event by E[max
Ξσ
ˆ (nj ) ,min Ξσ
ˆ (nj+1 ) ]

.
{τξ0 < τξˆ < τξ1 < · · · < τξm < τξ } ∈ Fτmin Ξ
1
σ
ˆ (nj+1 )
0
Theorem 4 implies that

σ
P[E[max
|Fmax Ξ
]
Ξσ
σ
ˆ (nj )
ˆ (nj ) ,min Ξσ
ˆ (nj+1 ) ]

≤

α
C m 4− 4 (se0 (j+1)−1)

nj+1 −1

Pσˆ (n) .
n=nj +1

Varying j = 0, 1, . . . , Me0 − 1 and using inequality (2.14), we see that

P[E σ ]

≤

−(α/4)
C |I| 4

Me 0
j=1 se0 (j)−1

Pi .
i∈I

If we use inequality (2.15) and |I| =

e Me

+ 1, we can deduce that the right hand side is

bounded above by
C |E| C

−(α/4)
e∈E Me 4

Me 0
j=1 se0 (j)

Pe .
e∈E

Recall that this estimate was based on a fixed e0 ∈ E, but the left hand side does not depend
on this choice. Taking the geometric average with respect to e0 ∈ E, we can see that

P[E σ ]

≤

C |E| C

− α
e∈E Me 4 4|E|

e∈E

Me s (j)
j=1 e

Pe .
e∈E

47

Note that the finer partition I and associated terms Me , se (j) are dependent on our initial
choice of order σ. Using the fact that E = ∪E σ and the above inequality, we get

P[E] ≤








C |E|

|S(Me ,(se (j))) |C
e
Me ;(se (j))M
j=1

− α
e∈E Me 4 4|E|

e∈E

Me s (j) 
j=1 e 




Pe ,
e∈E

e∈E

(2.16)
where

S(Me ,se (j))) = {σ ∈ S : Meσ = Me , sσe (j) = se (j), for 0 ≤ j ≤ Me , e ∈ E}

e
and the first sum in inequality (2.16) is over all possible Me ; (se (j))M
j=1

e∈E

. That is, for

each e, all possible Me ≥ 0 and possible orderings 0 = se (0) < se (1) < · · · < se (Me ) ≤ Ne .
Recall that Ne = |Ξe | − 1 is fixed with the initial partition E. To finish the proof of the
theorem, it suffices to show that the large term in the parenthesis in (2.16) can be bounded
above by some finite constant depending only on |E| and κ.
We claim that
|S(Me , se (j))| ≤ |E| e∈E Me +1 .

(2.17)

Notice that the pair (Me , se (j)) completely determines the partition {Ξi }i∈I σ , and the partition σ can be recovered from the induced partition σ
ˆ : {1, . . . , |I σ |} → I σ . This is because
the order that circles in any given Ξi are visited is predetermined by the ordering on Ξ, and
so moving from σ
ˆ to σ requires no new information. Next, we claim that σ
ˆ is determined
by knowing eσˆ (n) , for each 1 ≤ n ≤ |I σ |. If eσˆ (n) = e0 , then σ
ˆ (n) = (e0 , j0 ) where j0
can be determined by j0 = min{0 ≤ j ≤ Me0 : (e0 , j) : (e0 , j) ∈
/σ
ˆ (m), m < n}. There are
|I σ | =

ˆ (n) ,
e∈E (Me +1) terms to determine for eσ

48

each of which has at most |E| possibilities,

which proves (2.17). Using this to estimate the constant in (2.16), we see that

α

−
|S(Me ,(se (j))) |C e∈E Me 4 4|E|
e
Me ;(se (j))M
j=1

e∈E
α

−
|E| e∈E (Me +1) C e∈E Me 4 4|E|

≤
e
Me ;(se (j))M
j=1

Me s (j)
j=1 e

− α
(C|E|)Me 4 4|E|
e
Me ;(se (j))M
j=1
Ne

= |E||E|

Me s (j)
j=1 e

e∈E

e∈E

|E||E|

=

Me s (j)
j=1 e

e∈E

e∈E
e∈E
Me s (j)
j=1 e .

α

−
4 4|E|

(C|E|)Me

e∈E Me =1

0=se (0)<···<se (Me )≤Ne

This can be bounded above by removing the upper bound on the variable Me = M , and
letting the variables se (j) = s(j) start at its lowest possible point j and adding with no
upper bound, giving the inquality
∞

≤

∞

∞

|E||E|

M

e∈E M =0

∞

≤

M

∞

M

(C|E|)

∞


= |E||E| 
M =0

M


C|E|


− α s(j)
4 4|E|

j=1 s(j)=j

e∈E M =0



M s(j)
j=1

s(M )=M

s(1)=1

|E||E|

α

−
4 4|E|

···

(C|E|)

− α
1 − 4 4|E|



|E|
− α M (M +1) 
4 8|E|


.

Note that this bound only depends on |E| and κ. It is finite because inside the exponent it
has the form

2

an b−n < ∞ for some b > 1.

49

2.6

Main theorems

The strategy for proving Theorem 1 is to construct a family of circles Ξ and a partition
{Ξe }e∈E satisfying the hypothesis of Theorem 5 from the discs {|z − zj | ≤ rj } and the
distances lj . The constant given will depend on the size of the partition E, but then it can
be shown that that |E| can be bounded above in a way which depends only on the number
of points n.
Proof of Theorem 1. We can assume without loss of generality that for each j = 1, . . . , n,
h
the radius rj satisfies rj = lj /4 j for some integer hj ≥ 1. This is because the ratio must
−hj −1

satisfy 4

−hj

≤ rj /lj ≤ 4

for some integer hj , and any increase of rj by at most a

factor of 4 only affects the constant for each term j. Moreover, if hj ≤ 0, then the j-th term
in the inequality (2.1) is equal to 1, so we can assume that that hj ≥ 1.
Construct Ξ: for each 1 ≤ j ≤ n, construct a sequence of circles by

lj
ξjs = {|z − zj | = s }, for 1 ≤ s ≤ hj .
4
The family of circles {ξjs }j,s may not be disjoint, so we may have to remove some circles.
For any fixed k ≤ n, let Dk = {|z − zk | ≤ lk /4}, which is a closed disc containing all of the
circles centered at zk . For j < k ≤ n, define Ij,k = {ξjs : 1 ≤ s ≤ hj , and ξjs ∩ Dk = ∅}.
These are the circles centered at zj which intersect Dk if zk is closer to zj than zj is to
{0, 1, z1 , . . . , zj−1 }.
Define the family

Ξ = {ξjs : 1 ≤ j ≤ hj , 1 ≤ s ≤ hj }\ ∪1≤j<k≤n Ij,k .

50

Then Ξ is composed of mutually disjoint circles by construction. If dist(γ, zj ) ≤ rj , then γ
intersects each ξjs for 1 ≤ s ≤ hj , and therefore

P[∩n
j=1 {dist(zj , γ) ≤ rj }] ≤ P[∩ξ∈Ξ {γ ∩ ξ = ∅}].

(2.18)

In order to apply Theorem 5, we need to partition Ξ.
Partitioning Ξ: The family of circles already has a partition {Ξj }n
j=1 , where Ξj is the
set of circles ξ ∈ Ξ with center zj , but this may not be sufficiently fine to satisfy Theorem 5.
For example, it may be that there are circles centered at z3 which lie between circles centered
at z2 . To make a finer partition, for each Ξj , we will construct a graph Gj whose connected
components are the uninterrupted circles. Being more precise, the vertex set of Gj is Ξj , and
ξ1 , ξ2 ∈ Ξj are connected by an edge if, assuming ξ1 < ξ2 in the ordering on Ξ introduced
in the last section, the radii satisfy radξ1 = 4radξ2 , and the open annulus between ξ1 and
ξ2 contains no other circles from Ξ. Let Ej denote the set of connected components in Ξj .
Then each Ξj can be partitioned into {Ξe }e∈Ej , where Ξe is the vertex set of e ∈ Ej . The
circles in Ξe , for e ∈ Ej , are all concentric circles with center zj whose radii form a geometric
sequence with common ratio 1/4, and the closed annuli Ae are mutually disjoint.
By construction, we see that for any j < k and e ∈ Ej , the annulus Ae does not intersect
Dk , which contains every Ae for e ∈ Ek . Thus, if we let E = ∪n
j=1 Ej , Theorem 5 implies
that we can bound (2.18) by
n

P[∩ξ∈E {γ ∩ ξ = ∅}] ≤ C|E|
j=1 e∈Ej

Pyj (re )
Pyj (Re )

,

(2.19)

where re =rad(max{ξ ∈ Ee }) and Re =rad(min{ξ ∈ Ee }), where again the ordering for

51

the maximum and minimum is in terms of crossing time. It suffices now to show that |E|
Pyj (re )
is comparable to some value depending on n, and that e∈E
is comparable to
j Py (R )
e
j

Pyj (rj )
Pyj (Rj )

, where Rj = lj /4.

Bounding |E|: First, we observe a useful fact. If 1 ≤ j < k ≤ n, observe that
maxz∈D {|z − zj |}
k

minz∈D {|z − zj |}
k

=

|zj − zk | + lk /4
5
≤ < 4.
|zj − zk | − lk /4
3

(2.20)

To check the inequality, note that f (x) = (a + x)/(a − x) is an increasing function of x for
any a, and lk ≤ |zj − zk | by the definition of lk . Then |zj − zk | cancels in the fraction,
leaving 5/3. This inequality has the following two consequences:
a) If 1 ≤ j < k ≤ n, then |Ij,k | ≤ 1.
lj
b) If 1 ≤ j < k ≤ n, then ∪ξ∈Ξ ξ ⊂ Dk intersects at most 2 annuli { r ≤ |z − zj | ≤
k
4
lj
}.
4r−1
These follow readily from the definitions and (2.20), but full details are provided in Section
2.7.
These consequences can be used to bound |Ej |. Recall that |Ej | is the number of connected
components made from Ξj in the graph Gj , which comes from the simple graph Ξj where
sequential circles are connected. In order to create a new component, either a vertex or an
edge has to be removed. Consequence a) says that for each k > j, the number of vertices
removed is at most 1. Consequence b) says that for each k > j, the number of edges removed
is at most 2. Thus, for each 1 ≤ j ≤ n, the number of components created is bounded above

52

by

k>j (1 + 2)

= 3(n − j). It follows that |Ej | ≤ 1 + 3(n − j). Summing over j yields
n

|E| =

n

|Ej | ≤
j=1

1 + 3(n − j) = n +
j=1

3n(n − 1)
.
2

This completes the proof that C|E| can be bounded above by some constant Cn < ∞
depending only on n and κ.
Final estimate: To finish the proof, we need to show that
Pyj (re )
e∈Ej

Pyj (Re )

≤C

Pyj (rj )
Pyj (Rj )

.

We introduce some new notation. For any annulus A = {z : r ≤ |z − z0 | ≤ R}, where
y0 = 1 − |z0 | for z0 ∈ D fixed, let P (A) = Py0 (r)/Py0 (R). For 1 ≤ s ≤ hj , let Aj,s =
{lj /4s ≤ |z − zj | ≤ lj /4s−1 }, and let Sj = {s ∈ {1, . . . , hj } : Aj,s ⊂ ∪e∈Ξj Ae }. Using this
new notation, what we want to show is that

Pyj (re )
e∈Ej

Pyj (Re )

P (Aj,s ) ≤ Cn

=
s∈Sj

Pyj (rj )
Pyj (Rj )

hj

= Cn

P (Aj,s ).
s=1

By the estimate in Lemma 1, for each e ∈ Ej , we have Pyj (re )/Pyj (Re ) ≥ (re /Re )α =
4−α . Let Sjc = {1, . . . , hj }\Sj . Then

P (Aj,s ) =
s∈Sj

hj
s=1 P (Aj,s )
s∈Sjc P (Aj,s )

≤

Pyj (rj ) α|S c |
4 j.
Pyj (Rj )

We need to estimate |Sjc |. If s ∈ Sjc , then either s = 1 or there is some k > j so that
Dk ∩ Aj,s = ∅. By consequence b) of (2.20), for each k > j, this happens at most twice for

53

each k > j. Therefore, |Sjc | ≤ 1 +

k>j

2 = 1 + 2(n − j). By equation (2.19),
n

P[∩ξ∈Ξ {γ ∩ ξ = ∅}] ≤ C(n+(3/2)n(n−1))
j=1

Pyj (rj )
Pyj (Rj )

4α(1+2(n−j))

= Cn

2
4α(n )

n
j=1

Pyj (rj )
Pyj (Rj )

.

Using Theorem 1, the analogous estimate for whole-plane SLE can be proven. The
strategy will be to appeal to reversibility, reduce to the radial case, and then carefully use
the Growth Theorem to show that the estimates from the radial case are comparable to the
desired upper bound.
Proof of Theorem 2. By the reversibility of whole-plane SLEκ for κ ≤ 8 ([40],[21]), the
desired probability is the same as the corresponding probability when γ ∗ is a whole-plane
SLEκ from ∞ to 0, which we will assume for the rest of the proof. For any stopping time T ,
the path t → γ ∗ (T + t), conditioned on γ ∗ (−∞, T ), is a radial SLEκ path in C\γ ∗ (−∞, T ]
from γ ∗ (T ) to 0.
For z1 , . . . , zn ∈ C\{0}, let z0 = 0, and define R = max{|zj −zk | : 0 ≤ j, k ≤ n}. Assume
without loss of generality that rk ≤ lk for each k. Otherwise, the corresponding factor on
the right hand side of the estimate is 1. Let T = inf{t > −∞ : |γ ∗ (t)| = 4R}, which is
finite a.s. since the radial SLEκ tip converges to the target point with probability 1. Let
φ be a conformal map from C\γ ∗ (−∞, T ] to D which fixes 0 and sends γ ∗ (T ) to 1. Then,
conditioned on γ ∗ (−∞, T ], the path γ := φ(γ ∗ (T + ·)) is a radial SLEκ trace from 0 to 1.
Therefore,

∗
∗
∗
P[∩n
k=1 {dist(zk , γ ) < rk }|γ (−∞, T ]] = P[∩k {γ ∩ φ(B(zk , rk )) = ∅}|γ (−∞, T ]]. (2.21)

54

By the Schwarz lemma, for each k, zkT := φ(zk ) satisfies |zkT | ≤ 1/4, since |zk | ≤ R
and φ takes 4RD into D. Let rkT = max{|φ(z) − φ(zk )| : z ∈ ∂B(zk , rk )}, and define
lkT = min{|zk − 1|, |zkT − zjT | : 0 ≤ j < k}. Since |zkT | ≤ 1/4, lkT is not |zkT − 1| for any k,
and lkT ≤ 1/2 < 3/4 ≤ 1 − |zkT | := ykT . Therefore,
P T (rkT )
y
k

=
P T (lkT )
y
k

rkT
lkT

for each k = 1, . . . , n. Theorem 1 and (2.6) imply that (2.21) is bounded above by
n
T
P[∩n
k=1 {dist(zk , γ)

≤

rkT }|γ ∗ (−∞, T ]]

≤C
k=1

rkT
lkT

2−d

.

(2.22)

It suffices to show that the quotients rkT /lkT are uniformly comparable to rk /lk for each k.
For each k, we will apply the Growth theorem on the ball B(zk , 3R), since φ is conformal
on B(0, 4R). There is a small subtlety, in that φ may not preserve the order of z1 , . . . , zn .
That is, for a given k, if k ∗ ∈ {0, . . . , k − 1} so that lk = |zk − zk∗ |, it may be that
lkT = |zkT − zjT | for some j = k ∗ . This will only affect our estimate by a constant, since if we
define ˜lk = |zkT − zkT∗ |, we have by (2.6)

˜lk
|zk − zk∗ |
≤
·
|zk − zj |
lkT

|z −z |
1 + k3R j

2

(1 + (1/3))2
≤
1
·
= C.
2
(1 − 1/3)2
|zk −zk ∗ |
1 − 3R

The second inequality follows by the choice of k ∗ , and since |zk − zj | ≤ R for all 0 ≤
j, k ≤ n. Therefore, for each k, we have rkT /lkT ≤ C(rkT /l˜k ). By applying (2.6) to zk and

55

w ∈ ∂B(zk , rk ) with rkT = |φ(zk ) − φ(w)|, and to lk = |zk − zk∗ |, we get
lk
1 + 3R

rkT

2

r
rk (1 + (1/3))2
r
≤ k·
≤
= C k.
2
2
˜lk
rk
lk
lk (1 − 1/3)
lk
1 − 3R

Therefore, we have rkT /lkT ≤ Crk /lk uniformly.

Proof of Theorem 3. First, we prove the theorem for radial SLE. Recall that we defined
Contd (E; r) = rd−2 Area{z ∈ C : dist(z, E) < r} for any r > 0. Fixing r > 0, we have

E[Contd (γ; r)n ] = E[rn(d−2) (Area{z ∈ D : dist(z, γ) < r})n ]
n

= rn(d−2) E

I{z:dist(z,γ)<r} (z)dA(z)
D

=

Dn

rn(d−2) P [dist(z1 , γ) < r, . . . , dist(zn , γ) < r] dA(z1 ) . . . dA(zn ).

The last equality holds by Fubini’s theorem, since all of the terms are positive. By Theorem
1, we have

n

rn(d−2) P[dist(zk , γ)

< r, ∀k = 1, . . . , n] ≤

rn(d−2) C

n

Pyk (r ∧ lk )

k=1

Pyk (lk )

.

If r ≤ li1 , . . . , lik and r is greater than the rest, then the above expression is
k

=

rn(d−2) C

n

Pyi (r)
j

Pyi (lij )
j=1
j

k

≤ Cn

rn(d−2)

r2−d

2−d
j=1 lij

n

ljd−2 .

≤ Cn
j=1

To explain the last inequality, there are two cases. If r < lj , the numerator cancels with

56

one of the rd−2 ’s outside of the product. If r > lj , then one of the rd−2 ’s from the outside
product is added to the product, and is smaller than ljd−2 .
Thus, if f (z1 , . . . , zn ) =

n
d−2
k=1 min{|zk |, |zk − 1|, |zk − z1 |, . . . , |zk − zk−1 |}

and r > 0,

then
E[Contd (γ; r)n ] ≤ Cn

Dn

f (z1 , . . . , zn )dA(z1 ) . . . dA(zn ).

By Fatou’s Lemma,

E[Contd (γ)n ] ≤ lim inf E[Contd (γ; r)n ] ≤ Cn
r→0

Dn

f (z1 , . . . , zn )dA(z1 ) . . . dA(zn ).

If we can show that f is integrable, then we are done.
Fix k = 1, . . . , n, and let z1 , . . . zk−1 ∈ D be arbitrary. Let z−1 = 0 and z0 = 1. Then
for any k,
k−1

min{|zk |, |zk − 1|, |zk − z1 |, . . . , |zk − zk−1 |}d−2 dA(zk )

|zk − zj |d−2 dA(zk ).

≤
j=−1 D

D

Note that for each j, D − zj ⊂ 2D, and so the above inequality satisfies
2

|z|d−2 dA(z) = (k + 1)2π

≤ (k + 1)
2D

rd−1 dr < ∞.

0

By Fubini’s theorem, it follows that Dn f dA . . . dA < ∞, and so we are done in the radial
case.
The proof for whole-plane SLEκ follows similarly. We start off by writing

E[Contd (γ ∗ ∩D); r)n ] =

Dn

rn(d−2) P[dist(z1 , γ ∗ ) < r, . . . , dist(zn , γ ∗ ) < r]dA(z1 ) . . . dA(zn ).

57

The function f (z1 , . . . , zn ) excludes the distance from 1 in the whole-plane case. That is,
n

min{|zk |, |zk − z1 |, . . . , |zk − zk−1 |}d−2 .

f (z1 , . . . , zn ) =
k=1

Then
E[Contd (γ ∩ D)n ] ≤

Dn

f (z1 , . . . , zn )dA(z1 ) . . . dA(zn ).

If D ⊂ RD for R < ∞, then D − zj ⊂ 2RD for each zj ∈ D, so we can perform the same
bound as in the radial case, except integrating over 2RD rather than 2D.

2.7

Technical lemmas

The following lemma is proven in [23], and serves as a technical lemma to prove that a certain
random variable is a stopping time.
Lemma 11. Let D ⊂ C be a simply connected domain, and let ρ be a crosscut in D. Let
w0 , w1 , and w∞ be connected subsets or prime ends of D such that D\ρ is a neighborhood
of all of them in D. Suppose that ρ disconnects w0 from w∞ in D. Let γ(t), 0 ≤ t ≤ T, be a
continuous curve in D with γ(0) ∈ ∂D. Suppose for 0 ≤ t < T, D\γ[0, t] is a neighborhood
of w0 , w1 , and w∞ in D, and w0 , w1 ⊂ Dt := D(γ[0, t]; w∞ ). For 0 ≤ t < T , let ρt be
the first subcrosscut of ρ in Dt that disconnects w0 from w∞ as given by Lemma 2. For
0 ≤ t < T , let f (t) = 1 if w1 ∈ Dt (ρt ; w∞ ), and f (t) = 0 if w1 ∈ Dt∗ (ρt ; w∞ ). Then f is
right continuous on [0, T ), and left continuous at those t0 ∈ (0, T ) such that γ(t0 ) is not an
end point of ρt0 .
We can provide the complete rigorous details for some of the steps in the main theorems,
whose arguments would clutter the presentation of the proofs.
58

Proof of (2.10) in Theorem 4: First, I need to show that, in the event E, if t ≤ τj , then
D∗ (ξj ) is either contained in Dt (ρt ) or Dt∗ (ρt ) for j = 1, . . . , m. If t ≤ τj , and E occurs, then
ξj cannot be swallowed by the curve, since γ still needs to intersect ξj . Therefore, it must
be that ξj ⊂ Dt . Moreover, since D(ξj ) is disjoint from the curve ρ, it cannot be that the
subcrosscut ρt goes through D(ξj ). It follows that either D(ξj ) ⊂ Dt (ρt ) or D(ξj ) ⊂ Dt∗ (ρt ).
Similarly, in the event E, it must be that ξ0 ⊂ Dτm .
Let I be totally ordered by (0, 1) < (1, 1) < (1, 2) < (2, 2) < · · · < (m, m) < (m, m + 1).
Define a family of events by
Ei = E\ ∪i >i Ai .
Using a reverse induction argument, we can show that

Ei ⊂ {D∗ (ξi2 ) ⊂ Dτ∗ (ρτi )}
i1
1

for all i = (i1 , i2 ) = (m, m + 1). Note that E(m,m+1) = ∅. For the base case of the induction,
we consider

E(m,m) = E\A(m,m+1) = E ∩ {T ≤ τm } ∪ {Dτm (ξm ) ⊂ Dτm (ρτ ) } .
m

Note that E ∩ {T ≤ τm } = ∅, and the preceding paragraph shows that E ∩ {Dτm (ξm ) ⊂
Dτm (ρτm )} is contained in {Dτm (ξm ) ⊂ Dτ∗m (ρτm )}. This proves the base case.
To complete the induction, assume that E(j,j) ⊂ {D∗ (ξj ) ⊂ Dτ∗ (ρτj )}. Then
j

E(j−1,j) = E(j,j) ∩ Ac(j,j) ⊂

59

⊂ {D∗ (ξj ) ⊂ Dτ∗ (ρτj )} ∩ {D∗ (ξj ) ⊂ Dτ∗ (ρτj−1 )} ∪ {D∗ (ξj ) ⊂ Dτj (ρτj )} .
j
j−1
The intersection with the right hand side is empty, and so we get that E(j−1,j) is contained
in {D∗ (ξj ) ⊂ Dτ∗

j−1

(ρτj−1 )}. This completes the proof that the (j, j) case implies the

(j − 1, j) case. The argument for (j, j + 1) implies (j, j) is identical, and so the the induction
is complete.
The final step of the above induction sequence implies that

E\ ∪i>(0,1) Ai = E(0,1) ⊂ {D∗ (ξ1 ) ⊂ Dτ∗0 (ρτ0 )} ⊂ A(0,1) ,

from which claim (2.10) follows.

Proof of (2.13) in Theorem 4. Let x = Rj /rj , then N = ln(x) . We are trying to show

xα/N

1 − x−α/2
1 − x−α/2N

≤

eα
1 − e−α/4

.

If 1 < x ≤ e, then N = 1, and the left hand side is equal to

xα

1 − x−α/2
1 − x−α/2

= xα ≤ eα ≤

eα
1 − e−α/4

.

Suppose that x > e, in which case N ≤ ln(x) + 1, and so 1 − x−α/2(ln(x)+1) ≤ 1 − x−α/2N .
The function
1 − x−α/2(ln(x)+1)

=1−e

−(α/2)

ln(x)
ln(x)+1

is increasing, and so is bounded from below for x ∈ (e, ∞) at x = e, which yields the bound

60

1 − e−α/4 . Therefore, we get

1
1 − x−α/2N

≤

1
1 − e−α/4

.

Since ln(x) ≤ N = 0, we get that x1/N ≤ x1/ ln(x) = e, and so xα/N ≤ eα . Combining
this with the above inequality completes the proof.

Proof of (2.15) in Theorem 5. In the total order on Ξe , max{Ξ( e, j)} < min{Ξ(e,j+1) }, and
there are no circles between them by construction. By the assumption that sequential radii
in Ξe form a geometric sequence, we know that Rmax{Ξ
= 4Rmin{Ξ
Therefore,
(e,j) }
(e,j+1) }
expanding the product gives
Me

P(e,j) =
j=0

Pe (re )
Pe (Re )

Me −1
j=0

Me −1

Pye (Rmax{Ξ

)
(e,j) }

Pye (R{min{Ξ

(e,j+1) }}

)

≤ Pe




j=0

Rmax{Ξ

(e,j) }

Rmin{Ξ

(e,j+1) }

α
 = Pe 4αMe ,

which proves claim (2.15). Note that the above inequality follows from applying Lemma 1
to each term in the product.

Proof of consequences a) and b) in Theorem 1: To prove a), suppose that it is false. Then
rad(ξj1 )
1
2
there are distinct ξj , ξj ∈ Ij,k with
≥ 4. By both being in Ij,k , there is a z ∈ ξj1 ∩Dk
rad(ξj2 )
and a w ∈ ξj2 ∩ Dk . Then |z − zj |/|w − zj | ≥ 4. Taking the maximum over all z ∈ Dk
then implies that maxz∈D {|z − zj |}/|w − zj | ≥ 4. Taking the minimum of this in w ∈ Dk
k

contradicts (2.20).
The proof of b) follows similarly. Suppose b) is false. Then there exist 2 ≤ r1 < r2 <

61

r3 ≤ hj and w1 , w2 , w3 ∈ Dk such that
lj
lj
lj
lj
lj
lj
≤ |w3 − zj | ≤ r −1 ≤ r ≤ |w2 − zj | ≤ r −1 ≤ r ≤ |w1 − zj | ≤ r −1 .
r
43
42
41
43
42
41

From this series of inequalities, we get
lj /4r1
|w1 − zj |
= 4r3 −1−r1 ≥ 4.
≥
|w3 − zj |
lj /4r3 −1
This contradicts (2.20) in the same way as in a), proving b).

62

Chapter 3
Decomposition of backward SLE in
the capacity parametrization

3.1

Introduction

In some cases, the reverse flow of the Loewner equation is easier to study and can be used
to answer questions about the regular, or forward, SLE process. Analysis of the reverse
flow was used to show existence of the SLEκ trace for κ = 8 [27]. In [8], a multifractal
analysis is used to study moments for the backward SLE flow, which is used to provide a
new proof of the Hausdorff dimension of an SLE path. The reversibility of the welding [26]
has been used to study ergodic properties of the tip of a forward SLEκ in [39]. BSLE and
the conformal welding have been coupled with the Gaussian free field and what is called the
Liouville quantum zipper [33], where the welding of the real line onto the backward SLEκ
traces is seen as the conformal image of gluing random surfaces together. Recently [4], this
connection to the Gaussian free field was used to provide a new proof of the existence of the
forward SLE trace.
In this chapter, we construct a family of functions Ga,b (x, y) with which we use the
a,b

Girsanov theorem to construct a new measure Px,y under which φ(x) = y almost surely.
That is, we condition the process so that the graph of the welding function passes through

63

the point (x, y). In order to do this, we need the notion of a BSLE process with force points,
which are introduced in [26].
A process (ft ) which solves the chordal Loewner equation with driving function λ which
satisfies the SDE:

dλt =

√

κdBt +

adt
bdt
+
,
ft (q1 ) − λt ft (q2 ) − λt

λ0 = x0

is called a BSLEκ (a, b) process started from (x0 ; q1 , q2 ), where a, b ∈ R are weights at the
force points q1 , q2 ∈ R. This definition can be extended to include more than two force
points, which can also be placed in the interior of H, but this is the amount of generality we
will need. This process can be constructed by applying the Girsanov theorem, which will be
discussed in a later section, and therefore for κ ≤ 4, the backward SLE process with force
points also generates a conformal welding on R+ .
For any set U contained in the first quadrant, we then average the measures of paths
a,b

passing through U by integrating Px,y against IU (x, y)Ga,b (x, y). For κ ∈ (0, 4), and for
a = b = −4, we then prove a decomposition theorem relating this particular case to the
amount of capacity time that the graph of φ spends in U . In order to establish these results,
we review the framework established in [37] to study processes with a random lifetime.
For all κ ≤ 4, and for a = b = −κ − 4, we also show that the Green’s function Ga,b (x, y)
can be realized as the normalized probability the two points x, y are welded together. This
is similar to how the Green’s function for the forward SLEκ process captures the normalized
probability that the path passes through a given point.

64

3.2

Processes with a random lifetime

For this paper, we will need to review the framework introduced in [37] to study stochastic
processes with a random lifetime. We will need to introduce the notation which will be used
in this paper, and several propositions will be stated without proof. For complete details,
the reader can refer to [37].
Define the space
Σ = ∪0<T ≤∞ C[0, T ),
where C[0, T ) is the set of real valued functions which are continuous on [0, T ). Then the law
of Brownian motion is a probability measure on Σ. We will have to define a few operations
on this space.
• Lifetime: First, the lifetime of a function f ∈ Σ is given by Tf = sup{t > 0 : f ∈
C[0, t)}.
• Killing: For 0 < τ ≤ ∞, define Kτ : Σ → Σ by Kτ (f ) = f|[0,τ ) , where τf =
f
min{τ, Tf }.
• Continuing: This is an operation which glues two functions together. Define subspaces
of Σ by Σ⊕ = {f ∈ Σ : Tf < ∞, f (T − ) := limt→T − f (t) ∈ R} and Σ⊕ = {f ∈ Σ :
f
f (0) = 0}. Then we define ⊕ : Σ⊕ × Σ⊕ → Σ by

f ⊕ g(t) =





f (t),

0 ≤ t < Tf

.




f (T − ) + g(t − Tf ), Tf ≤ t < Tf + Tg
f

ˆ : Σ⊕ × Σ⊕ → Σ × [0, ∞) by f ⊕g
ˆ = (f ⊕ g, Tf ).
• Time marked continuation: Define ⊕
65

ˆ records the information of where the first function ends and the second begins.
Then ⊕
Probability measures on Σ will be the laws of the random driving function for the backward Loewner process. To talk about measures, we need to create a sigma algebra on Σ.
First, for t < ∞, define a filtration by

Ft = σ {f ∈ Σ : s < Tf , f (s) ∈ U }, s ≤ t, U ⊂ R is measurable .

Then define F = ∨0≤t<∞ Ft . Then the operations above are all measurable on (Σ, F). Given
any two measures µ, ν on (Σ, F) let µ ⊗ ν denote the product measure. We then define the
following measures:
• µ ⊕ ν is defined by the pushforward measure ⊕∗ (µ ⊗ ν) on Σ.
ˆ is the pushforward measure ⊕
ˆ ∗ (µ ⊗ ν) defined on Σ × [0, ∞).
• µ⊕ν
We will also have to work with random measures on different spaces, which are called
probability kernels. More precisely, suppose (U, U) and (V, V) are measurable spaces. A
kernel from (U, U) to (V, V) is a map ν : (U, V) → [0, ∞) such for each u ∈ U , ν(u, ·) : V →
[0, ∞) is a measure and for each E ∈ V, the function ν(·, E) : U → [0, ∞) is U-measurable.
If µ is a measure on (U, U), then ν is called a µ-kernel if it is a kernel on the µ-completion
of (U, U). We say that ν is a finite µ-kernel if ν(u, V ) < ∞ for µ-a.s. u ∈ U, and we say
ν is a σ-finite µ-kernel if V = ∪∞
n=1 Fn such that for each n, for µ-a.s. u ∈ U , we have
ν(u, Fn ) < ∞.
Combining kernels with measures, we have the following operations for measures:
• If µ is a σ-finite measure on (U, U) and ν is a σ-finite µ-kernel from (U, U) to (V, V),

66

then we define µ ⊗ ν on U × V by

µ ⊗ ν(E × F ) =

ν(u, F )dµ(u).
E

In this case, µ · ν is the marginal measure on V given by µ · ν(F ) = µ ⊗ ν(U × F ).
• If ν is a σ-finite measure on V and µ is a σ-finite ν-kernel from (V, V) to (U, U), we
←
−
define µ ⊗ ν on U × V by

←
−
µ ⊗ ν(E × F ) =

µ(v, E)dν(v).
F

• If ν is a σ-finite µ-kernel from Σ to (0, ∞), define a measure Kν (µ) on Σ to be the
pushforward measure of µ ⊗ ν under the map K : Σ × (0, ∞) → Σ given by (f, r) →
Kr (f ).
To study the law of a random process over time, the notion of absolute continuity can be
extended to what we call local absolute continuity. For 0 ≤ t < ∞, let Σt = {f ∈ Σ : Tf > t}
be the set of functions defined at least until time t. Note that ∩t>0 Σt = C[0, ∞). For each
t > 0, let Ft ∩ Σt denote the restriction of Ft onto the subspace Σt . If µ, ν are measures
on (Σ, F), then we say that ν is locally absolutely continuous with respect to µ if, for every
t > 0, ν|Ft ∩Σt is absolutely continuous with respect to µ|Ft ∩Σt . We will use the notation
ν

µ to mean absolute continuity and ν

µ to mean local absolute continuity.

The following propositions are in [37], and will be stated without proof:
Proposition 2. Let µ be a measure on (Σ, F) which is σ-finite on F0 . Let (Υ, G) be a
measurable space. Let ν : Υ × F → [0, ∞] be such that for every v ∈ Υ, ν(v, ·) is a finite
measure on F that is locally absolutely continuous with respect to µ. Moreover, suppose that
67

the local Radon Nikodym derivatives are equal to (Mt (v, ·))t>0 , where Mt : Υ × Σ → [0, ∞)
is G × Ft measurable for every t ≥ 0. The ν is a kernel from (Υ, G) to (Σ, F). Moreover, if
ξ is a σ-finite measure on (Υ, G) such that µ-a.s., Υ Mt (v, ·)dξ(v) < ∞, then ξ · ν

µ, and

the local Radon-Nikodym derivatives are Υ Mt (v, ·)dξ(v) for 0 ≤ t < ∞.
Proposition 3. Let µ be a probability measure on (Σ, F). Let ξ be a µ-kernel from (Σ, F) to
(0, ∞) that satisfies Eµ [|ξ|] < ∞. Then Kξ (µ)

µ, and the local Radon-Nikodym derivatives

are Eµ [ξ((t, ∞))|Ft ] for 0 ≤ t < ∞.
Throughout this chapter, we will use Pκ to denote the measure on Σ which gives the law
√
of ( κBt )t>0 , where (Bt )t>0 is a standard one dimensional Brownian motion. Then Pκ is
supported on C[0, ∞) ∩ Σ⊕ . We will use Eκ to denote expectation under the measure Pκ ,
and FtB to be the completion of Ft under Pκ for 0 ≤ t.
We state two more propositions from [37] without proof. The first extends the Girsanov
theorem to a statement about local absolute continuity, and the second extends the strong
Markov property of Brownian motion.
Proposition 4. Suppose that (Xt )0≤t<T satisfies the FtB -adapted SDE:

dXt =

√

κdBt + σt dt,

0 ≤ t < T, and X0 = 0,

where T is a positive FtB -stopping time. Let Pκ,σ denote the law of X. Then Pκ,σ

Pκ .

Moreover, if (Mt )0≤<T is a FtB -adapted continuous local martingale that satisfies the SDE
σt
dMt = Mt √ dBt ,
κ

0 ≤ t < T0 , and M0 = 1,

68

(3.1)

then
dPκ,σ |Ft ∩Σt
dPκ |Ft ∩Σt

= I{T >t} Mt ,

0 ≤ t < ∞.

(3.2)

If (θ)t is an FtB -adapted increasing process, it induces a kernel from Σ to [0, ∞) defined by
dθ· (f ), which is the random measure induced by the monotonic function θt (f ) on t ∈ [0, ∞).
Proposition 5. Let (θt )0≤t<∞ be a right continuous increasing FtB -adapted process that
satisfies θ0 = θ0+ = 0 and Eκ [θ∞ ] < ∞. Then

ˆ κ = Pκ ⊗ dθ.
Kdθ (Pκ )⊕P

Thus, Kdθ ⊕ Pκ

(3.3)

Pκ , and θ∞ is the Radon-Nikodym derivative.

Define ΣL to be the set of λ ∈ Σ which generate a Loewner trace. By the existence of the
SLEκ trace for any κ > 0, we know that Pκ is supported in ΣL . Let ΣC = ∪0<t≤∞ CC [0, t),
where CC [0, t) is the set of complex valued continuous functions with domain [0, t). Define the
Loewner operator L : ΣL → ΣC to be the map which takes a driving function to its chordal
trace. Then the pushforward measure L∗ (Pκ ) is the law of the SLEκ curve. More generally,
if µ is any law supported in ΣL , then L∗ (µ) is the law of the associated Loewner curve.
We can also define the extended Loewner map Lˆ : {(λ, t) : λ ∈ ΣL , t < Tλ } → ΣC × H by
ˆ t) = (L(λ), L(λ)(t)). If ν is any measure on {(λ, t) : λ ∈ ΣL , t < Tλ }, then Lˆ generates
L(λ,
a pushforward measure Lˆ∗ (µ) on ΣC × H. This approach is taken in [37] to study chordal
SLEκ in both the capacity parametrization and natural parametrization.
Since our object of study is the conformal welding generated by the driving function,
we will consider a different operator. Let ΣW denote the functions in ΣL which generate a
continuous conformal welding under the backward Loewner equation. For κ ≤ 4, since the
69

Loewner traces are simple, it follows that Pκ is supported in ΣW . We will use Q1 to denote
the first quadrant [0, ∞) × [0, ∞). The welding operator W : ΣW → ΣC can be defined
by W(λ) = the welding curve Φ generated by λ. We can also define the extended welding
ˆ
operator W(λ,
t) = (Φ, Φ(t)) ∈ ΣC × Q1 . Then any law µ supported on ΣW generates a
pushforward measure W∗ (µ) on ΣC , and any law ν on {(λ, t) : λ ∈ ΣW , t < Tλ } generates a
ˆ ∗ (ν) on ΣC × Q1 .
pushforward measure W
Lemma 12. If κ ∈ (0, 4), then the welding operator W is injective on the support of Pκ .
˜ are (ft )t≥0 and (f˜t )t≥0
Proof. Suppose the backward Loewner processes generated by λ, λ
respectively, and assume they generate the same welding curve Φ. For a fixed t > 0, define
˜ t , and let Gt (z) = F˜t ◦F −1 (z), which maps H\{γt −λt }
Ft (z) = ft (z)−λt and F˜t (z) = f˜t (z)−λ
t
˜ t }, where (γt )t≥0 and (˜
onto H\{˜
γt − λ
γt )t≥0 are the families of backward Loewner traces
˜ respectively.
driven by λ and λ
For κ < 4, since γt − λt has the same law under Pκ as a forward SLEκ trace up to time
t, it was proven in [27] that γt − λt is the boundary of a H¨older domain. In [6], it was shown
that this condition is enough to make the image of the trace conformally removable. That
is, any conformal map defined on H\{γt − λt } which can be extended continuously to the
boundary can be conformally extended to H.
Suppose Φ(t) = (bt , at ), and assume −at ≤ −y < 0 < x ≤ bt with φ(x) = y. Note
that since both processes have the same welding process, they also have the same welding
function φ. Therefore, ft (x) = ft (−y) and f˜t (x) = f˜t (−y), and so Gt extends continuously
˜ t . Conformal removability then implies that Gt extends to a conformal
from γt − λt to γ˜t − λ

70

map from H onto H, and so it must be that

Gt (z) =

az + b
cz + d

for some a, b, c, d ∈ R with ad − bc = 1.
˜ t . This
Assuming Φ(t) = (bt , at ), we get ft (−at ) = ft (bt ) = λt and f˜t (−at ) = f˜t (bt ) = λ
implies that Gt (0) = 0, and so b = 0. Also, ft and f˜t each fix ∞, and so c = 0. Therefore,
˜ t , the scaling rule for half
Gt (z) = az for some constant a. Since Gt (γt − λt ) = γ˜t − λ
˜ t ) = 2t, and so a = 1 or
plane capacity implies that 2t = hcap(γt − λt ) = |a|2 hcap(˜
γt − λ
a = −1. Since ft and f˜t both have positive derivative at ∞, we have a = 1. Therefore,
˜ t for each t > 0. Since the reverse flow is uniquely determined by the driving
γt − λt = γ˜t − λ
˜
function, we get that λ = λ.

3.3

Glossary of notation for Ito’s formula calculations

We define some terms which will be used for Ito’s formula calculations for this chapter in
multiple sections.
• Fix x, y > 0, and let λt be a driving function for the backward Loewner equation (1.4).
Let (ft )t≥0 be the solution of (1.4) driven by λ. For z ∈ R, let τz be the lifetime of
(1.4) started from z. If x, y > 0, let τx,y = min{τx , τ−y }.
• For t ≤ τx , define Xt = Xt (x) = ft (x) − λt .
• For t ≤ τ−y , define Yt = Yt (y) = λt − ft (−y). Note that Xt and Yt are defined
specifically to be nonnegative for all t ≤ τx,y .
71

• These two formulas imply that

d(Xt + Yt )
−2
=
dt.
Xt + Yt
Xt Yt
• For t < τx,y , let Wt = Wt (x, y) =

(3.4)

Xt − Yt
. Then Wt ∈ (−1, 1) for each t < τx,y . If
Xt + Yt

τx = τ−y , then Wτx,y is well defined and is either 1 or −1.
• For t ≤ τx,y , define ut = ut (x, y) = 0t

κ
dr. By equation (3.4), this is equal to
Xr Yr

κ
u(t) = − ln
2

X t + Yt
x+y

.

(3.5)

The process ut will be used to define a random time change.
• For Z ∈ {X, Y, W }, we will use Zˆt = Zu−1 (t) .

3.4

Generalized Green’s functions

In this section, we will be doing Ito’s formula calculations using the framework in Section
3.3. For the first lemma, λt =

√

κBt is the driving function for BSLEκ . We will then do

calculations when λt is the driving function for the BSLEκ (a, b) process.
For κ ≤ 4, let (ft ) be the standard backward SLEκ process driven by λt =

√
κBt . For

x, y > 0 define Xt = Xt (x) = ft (x) − λt and Yt = Yt (−y) = λt − ft (−y) as in Section 3.3.
Ito’s formula then implies that
√
√
2
2
dt, and dYt = κdBt − dt.
dXt = − κdBt −
Xt
Yt

72

Note that ft (x) is decreasing in t and ft (−y) is increasing in t. Since λt is unbounded in
both directions, then τx = inf{t > 0 : Xt (x) = 0} < ∞ a.s. and τ−y = inf{t > 0 : Yt (y) =
0} < ∞ with probability 1, and Xt , Yt track the flow of x and −y towards 0. The stopping
time τx,y = τx ∧ τ−y stops the process as soon as either x or −y are absorbed by the welding.
What we want to do is weight the measure Pκ in such a way that τx,y = τx = τ−y almost
surely. That is, we want to change the probability measure so that φ(x) = y with probability
a,b

1. We will find a new family of measures Px,y on Σ which do this, and we call the process
an extended BSLEκ (a, b) process started from (0; x, −y).
a,b

Proposition 6. Fix a, b ∈ R. Then Mt

a

b

−
−
a,b
= Mt (x, y) := Xt κ Yt κ (Xt +Yt )γ ft (x)p ft (−y)q

is a local martingale if and only if

p=−

a(a + κ + 4)
,
4κ

q=−

b(b + κ + 4)
ab
, and γ = − .
4κ
2κ

In this case, the process (ft )t>0 weighted by this local martingale using the Girsanov theorem
is a backward SLEκ (a, b) process started from (0; x, −y).
Proof. For ease of notation, let α = −a/κ and β = −b/κ. This follows from an Ito’s formula
calculation, which will be included for completeness. Observe first that

∂t ft (z) =

2ft (z)
d
−2
∂t ft (z) =
·
(−f
(z))
=
.
t
dz
(ft (z) − λt )2
(ft (z) − λt )2

Therefore,
dft (z)r
df (z)
2r
=r t
=
dt.
r
ft (z)
ft (z)
(ft (z) − λt )2

73

Applying this at both x and −y gives
dft (x)p
dft (−y)q
2p
2q
=
dt
and
= 2 dt.
p
q
2
ft (x)
ft (−y)
Xt
Yt
Also, direct applications of Ito’s formula gives
√
dXtα
−α κ
h(α)
=
dBt +
dt,
α
Xt
Xt
Xt2

d(Xt + Yt )γ
−2γ
dt,
=
γ
(Xt + Yt )
Xt Yt

√
α κ
h(β)
=
dBt + 2 dt,
β
Yt
Yt
Yt
β

dYt

where h(t) = −2t +

t(t − 1)κ
. By the product rule, we get that
2
β

β

d Xtα , Yt
dXtα dYt
dMt
=
+ β +
β
Mt
Xtα
Yt
Xtα Yt
β

=

Notice that

d(Xt + Yt )γ dft (x)p dft (−y)q
+
+
+
(Xt + Yt )γ
ft (x)p
ft (−y)q
β

d Xtα , Yt
dXtα dYt
−2γ
2p
+
+
+
+ 2+
=
α
β
β
Xt
Xt Yt Xt
Yt
Xtα Yt
√
√
−α κ β κ
h(α) + 2p h(β) + 2q
+
+
−
dBt +
Xt
Yt
Xt2
Yt2

2q
Yt2

dt

αβκ + 2γ
Xt Yt

dt.

a
b
c
+
−
= 0 for all x, y, ∈ R if and only if a = b = c = 0. This occurs if
x2 y 2 xy

and only if p, q, and γ are chosen as indicated. Once these are chosen, we then get that

dMt
=
Mt

√
√
−α κ −β κ
+
Xt
Yt

dt.

Using this as a drift term and weighting the original measure to obtain a new probability

74

˜ the Girsanov theorem says that
measure P,

dλt =

√

κdBt =

√

˜t +
κdB

−ακ
−βκ
+
ft (x) − λt ft (−y) − λt

dt,

˜
˜t is a P-Brownian
where B
motion. This is precisely the definition of a backward SLEκ (−ακ, −βκ)
process started from (0; x, −y), where a = −ακ and b = −βκ.

The force points at x and −y serve as magnets. If a and b are chosen correctly, we can
force x and −y to be absorbed by the process at the same time, which would imply that
φ(x) = y. Along with Xt and Yt , define Wt as in Section 3.3 by Wt =

X t − Yt
, which is
X t + Yt

a process that stays in (−1, 1) for as long as it exists. It only fails to exist if τx = τ−y ,
where τz is the time at which z is swallowed by the BSLE traces. If this does not happen
(which is a.s. under Pκ ), then Wt is constant at ±1 after τx,y . The strategy is to perform
an appropriate random time change, under which W becomes a diffusion process. We can
then analyze the lifetime of this diffusion.
Lemma 13. A BSLEκ (a, b) process started from (0; x, −y) welds x and y together with
probability 1 if a, b ≤

−4 − κ
.
2

Proof. Consider a backward chordal SLE(κ; a, b) process started from (0; x, −y) with driving
function λ. Let Xt , Yt , Wt , and u(t) be defined as in Section 3.3. We proceed similarly to
Proposition 6. By definition,
√
dXt = − κdBt −

a
b
−
X t Yt

dt −

√
2
dt, and dYt = κdBt +
Xt

75

a
b
−
Xt Yt

dt −

2
dt.
Yt

Therefore,
√
dXt = − κBt −

a+2
b
−
Xt
Yt

dt, and dYt =

√

κdBt +

a
b+2
−
Xt
Yt

dt.

(3.6)

Simple Ito’s formula calculations show that

dXt + Yt
−2
dXt − Yt
2
2
=
dt, and
=
dt −
dλt ,
Xt + Yt
X t Yt
Xt − Yt
Xt Yt
Xt − Yt
and therefore
√
4
2
−2 κdBt
dWt
=
−
dλt =⇒ dWt =
+
Wt
Xt Yt Xt − Yt
Xt + Yt

Using the basic algebraic equality 2

4Wt
bXt − aYt
+2
X t Yt
(Xt + Yt )Xt Yt

dt.

x−y
bx − ay
= (b − a) + (b + a)
, this becomes
x+y
x+y

√
−2 κdBt
dWt =
+
Xt + Yt

4Wt
(b − a) + (b + a)Wt
+
Xt Yt
Xt Yt

dt.

κ
κ
dr = − ln((Xt + Yt )/(x + y)).Using u to perform a
Recall the function u(t) = 0t
X r Yr
2
random time change, we get

˜t = −
dW

˜
˜ 2 dB
˜t + (b − a) + (b + a)Wt dt,
1−W
t
κ

(3.7)

˜ t = W −1 . Anything satisfying equation (3.7) is
where B˜s is a Brownian motion and W
u (t)
associated with a radial Bessel process, and is studied in [39]. Thus, our process W is such
a process traversed at a random speed. Observe that uτx,y is the lifetime of (3.7). Also,
note that Xτx,y + Yτx,y = 0 if and only if uτx,y = ∞. According to the appendix in [39], the

76

lifetime of (3.7) is infinite if a, b ≤ −2 −

κ
κ
with probability 1. Hence, if a, b ≤ −2 − , then
2
2

Xτx,y + Yτx,y = 0, and thus x and y are welded together.
Proposition 6 and Lemma 13 motivate us to define the a, b-backward SLEκ Green’s
ab

b

a

function for a, b ≤ −2 − κ/2 by Ga,b (x, y) = x− κ y − κ (x + y)− 2κ , so that M a,b (x, y) =
Ga,b (Xt , Yt )ft (x)p ft (−y)q is the local martingale whose weighting gives the BSLEκ (a, b)
process starting from (0; x, −y). What we want to do now, however, is run the backward
SLEκ (a, b) process until it welds x and y together, and then revert the driving function back
to a standard Brownian motion after τx,y .
a,b

To make this precise, let a, b ≤ −2 − κ/2. Let Px,y on Σ be the measure of the driving
function for the BSLEκ (a, b) process starting from (0; x, −y) until time τx,y . Then τx,y < ∞
a,b

means that Px,y is supported on Σ⊕ . Since Pκ , the law of

√

κ times a Brownian motion, is

a,b

supported on Σ⊕ , that means we can define the measure Px,y ⊕ Pκ on ΣW , which we will call
the law of the driving function for an extended BSLEκ (a, b) process started from (0; x, −y).
a,b

Then W∗ (Px,y ⊕Pκ ) is the measure on ΣC giving the law of the extended BSLEκ (a, b) started
from (0; x, −y) welding curve. Under this measure, the welding curve passes through (x, y)
almost surely.
Our goal is to show that backward SLE can be decomposed into the average of weighted
extended BSLEκ (a, b) processes. Fix a measurable U ⊂ Q1 = [0, ∞) × [0, ∞), and observe
a,b

that the measure Px,y is a probability kernel from Q1 to (Σ, F). Define a new measure on
(Σ, F) by
a,b

PU =

a,b

U

Px,y Ga,b (x, y)dxdy.

We say that BSLEκ admits a BSLEκ (a, b) decomposition if for all measurable U ⊂ Q1

77

there is some increasing random process Θ = ΘU
t

t>0

so that

a,b ˆ
U
PU ⊕P
κ = Pκ ⊗ dΘ· .

Recall then that dΘU
· is a kernel from Σ to [0, ∞).
This is a backward analogue to the definition in [37], which defines what it means for SLEκ
to admit an SLEκ (ρ) decomposition. In [37], it is proven that SLEκ admits both an SLEκ (κ−
8) decomposition and an SLEκ (−8) decomposition. In the first case, this corresponds to
weighting the SLEκ curve against the natural parametrization, and in the latter the SLEκ
curve is weighted by the capacity parametrization. In the next section, we will show that
BSLEκ admits a BSLEκ (−4, −4) decomposition, and see that the corresponding weight is
capacity time.
a,b

We will need to study the distribution of the stopping time τx,y under Px,y .
a,b

Lemma 14. Under Px,y , the stopping time τx,y satisfies

τx,y =

(x + y)2 ∞ − 4 t
ˆ 2 )dt
e κ (1 − W
t
4κ
0

(3.8)

a,b
ˆ satisfies (3.7) for a, b ≤ −2 − κ/2.
Px,y a.s., where W

Proof. Recall the time change from before, where u(t) := 0t

κ
κ
ds = − ln((Xt +Yt )/(x+
Xs Ys
2

y)), and we will use the convention that Zˆt = Zu−1 (t) , for Z ∈ {X, Y, W }. We claim that
a,b

Px,y -almost surely,

∞

τx,y =
0

ˆ s Yˆs
X
ds.
κ

(3.9)

2
a,b
Recall that under Px,y ,we have τx = τy , and so u(τx,y ) = − ln(Xτ + Yτ ) = ∞ Px,y -almost
κ
78

surely, where τ = τx,y . We are using the fact that x and y are welded together here. Then
∞
0

u−1 (τ ) X
τ
τ X Y
ˆ s Yˆs
ˆ s Yˆs
(u−1 ) (s)
X
t t
ds =
u
(t)dt
=
1dt = τx,y .
ds
=
κ
κ (u−1 ) (u(u−1 (s)))
κ
u−1 (0)
0
0

Using W =

X −Y
4XY
ˆ t + Yˆt =
and the identity 1 − W 2 =
, and the fact that X
X +Y
(X + Y )2

2

(x + y)e− κ t , we can rewrite the integrand in equation (3.9) and obtain (3.8).

a,b
Define a function σ a,b (x, y; t) = Px,y [τx,y ≤ t]. By equation (3.8), we can conclude that

σ a,b (x, y; t) = σ a,b

x y
√ , √ ; 1 , for all t > 0.
t t

(3.10)

This holds because
(x + y)2 ∞ − 4 s
ˆ 2 )ds ≤ t
e κ (1 − W
s
4κ
0
if and only if

√
√
(x/ t + y/ t)2 ∞ − 4 s
ˆ 2 )ds ≤ 1,
e κ (1 − W
s
4κ
0

√
√
ˆ t )t>0 has the same starting point at (x, y) as at (x/ t, y/ t).
where the diffusion (W
It is worth observing that the local martingale obtained in Proposition 6 is a local martingale, but not a martingale.
Lemma 15. If a, b < 0, then

a,b

Eκ [Mt (x, y)] = Ga,b (x, y)Px,y (τ(x,y) > t) = xα y β (x + y)γ (1 − σ a,b (x, y; t)),

(3.11)

where α = −a/κ, β = −b/κ, and γ = −ab/(2κ). In particular, Mt is a local martingale, but

79

a strict supermartingale.
Proof. For this lemma, since x and y are fixed, we will let τ = τx,y . Define τn = inf{t :
Mt ≥ n}. Then for each n, Mt∧τn is a bounded local martingale, and hence a martingale.
Therefore,
Eκ [Mt ] = lim Eκ [Mt I{τn >t} ] = lim Eκ [Mt∧τn Iτn >t ]
n→∞

n→∞

a,b

a,b

= G(x, y) lim Px,y (τn > t) = G(x, y)Px,y (∪n {τn > t}) .
n→∞

a,b

a,b

We claim that Px,y (∪n {τn > t}) = Px,y (τ > t). It is clear that ∪n {τn > t} ⊂ {τ > t},
since τ is the lifetime of (Mt ). It suffices to show ∩n {τn ≤ t} ⊂ {τ ≤ t}. In the event
∩n {τn ≤ t}, Ms is unbounded for s ∈ [0, t]. It can easily be seen that if t ≤ τ , then
ft (x) − λt ≤ x − λt ≤ x + y, and λt − ft (−y) ≤ x + y similarly. It follows immediately that
β

Xtα Yt ≤ (x + y)α+β , and so this factor of Mt is bounded uniformly for fixed x, y. Therefore,
if Ms is unbounded on [0, t], then it must be that either (Xs + Ys )γ or fs (x)p fs (−y)q is
unbounded on [0, t]. In the first case, since γ = −ab/(2κ) < 0, this implies that Xs + Ys → 0
before time t. That is, τ(x,−y) ≤ t. In the latter case, the reverse Loewner equation implies

fs (x)p fs (−y)q = exp{

s

2q
2p
+ 2 dr}.
2
Yr
0 Xr

This term is unbounded only if p and q are nonnegative and Xs , Ys → 0 before time t.
Therefore, in either case, we have τ ≤ t.

80

3.5

Capacity parametrization

The goal of this section is to show that BSLEκ has a BSLEκ (−4, −4) decomposition, and
to show the relationship between this decomposition and the capacity time for the welding
−4,−4

curve. For this section, we will use Px,y := Px,y

for x, y > 0.

Lemma 16. 0∞ 0∞ Px,y [τx,y ≤ 1]dxdy ∈ (0, ∞).
This lemma states that the probability of two points which are far away being welded
together in a short time decays rapidly. By the scaling relation, what matters most is how
relatively far x is from 0 in comparison to y. This is reflected in the proof, where we perform
a change of variables from (x, y) to r(t, 1 − t) and perform the most work integrating the
single variable t.
Proof. By equation (3.8), we have Px,y -almost surely that

τx,y ≤

(x + y)2 ∞ − 4 t
(x + y)2
e κ dt =
.
4κ
16
0

Thus, if x + y ≤ 4, then

Px,y [τx,y ≤ 1] ≥ Px,y [τx,y ≤ (x + y)2 /16] = 1,

and therefore
∞
0

∞
0

Px,y [τx,y ≤ 1] ≥

1dxdy > 0.
x+y≤4

To use the scaling relation for BSLE to show that 0∞ 0∞ Px,y [τx,y ≤ 1]dxdy < ∞,
we perform a change of variables. Observe that the distribution of τx,y is the same as
x
. To simplify notation, we will write Pt = Pt,1−t and
(x + y)2 τt,1−t , where t =
x+y
81

τt = τt,1−t for 0 < t < 1. Performing the change of coordinates x = rt, y = r(1 − t), we get
that
∞
0

∞
0

∞

1

Px,y [τx,y ≤ 1]dxdy =

0

0

1
Pt [τt ≤ 2 ]rdrdt.
r

For any fixed t, observe that
∞
0

Pt [τt ≤ r−2 ]rdr =

∞
0

Pt

1
1
1
≥ r2 rdr = Et
,
τt
2
τt

where Et is the expectation with respect to the measure Pt . Thus, it suffices to show that
1
1
0 Et τ dt is finite.
t
For any small δ > 0, we can break up the integral as
1
0

Et

δ/2
1−δ/2
1
1
1
1
1
dt =
Et
dt +
Et
dt +
Et
dt.
t
τt
τt
τt
0
δ/2
1−δ/2

To complete the proof, we will produce a small δ > 0 so that:
a) There is a constant Cδ < ∞ so that if 0 < t ≤ δ/2,

Et

1
1
≤ Cδ Eδ/2
τt
τδ/2

< ∞.

(3.12)

b) For each t ∈ (δ/2, 1 − δ/2),

Et

1
1
1
≤ Eδ/2
+ E1−δ/2
.
τt
τδ/2
τ1−δ/2

(3.13)

Note that if (3.12) is proven, a symmetric argument proves a similar bound for t ∈ [1−δ/2, 1).
Once a correct δ can be chosen for (3.12), then (3.13) and both boundary estimates imply
82

that 01 Et [1/τt ] dt < ∞.
Note that t0 = x/(x + y) being near 0 is the same as starting the process Wt = (Xt −
Yt )/(Xt + Yt ) at w = (x − y)/(x + y) = 2t0 − 1 near −1, and t0 being near 1 corresponds to
starting the process Wt near 1. Under the measure Pt0 , equation (3.7) reduces to
ˆ 2 dBt + −8 W
ˆt = − 1 − W
ˆ t dt,
dW
t
κ

ˆ 0 = w.
W

(3.14)

We will use this diffusion to prove (3.13). Suppose δ > 0 is small, and fix any t∗ ∈ (δ/2, 1 −
δ/2). Let (Wt∗ )t>0 be (3.14) started at w = 2t∗ −1, (Wt1 )t>0 be (3.14) started at w = −1+δ,
and let (Wt2 )t>0 be (3.14) started at w = 1 − δ. Then for each t > 0,

j

1 − (Wt∗ )2 ≥ min{1 − (Wt )2 : j = 1, 2}.

Thus, by equation (3.8), we can see that there is a coupling so that τt∗ ≥ min{τδ/2 , τ1−δ/2 },
where τt∗ , τδ/2 , and τ1−δ/2 are defined on the same probability space, and each τw has the
distribution of τw under Pw for w ∈ {δ, 1 − δ/2}. Therefore,
1
1
1
1
1
≤ max{
,
}≤
+
.
τt∗
τδ/2 τ1−δ/2
τδ/2 τ1−δ/2
Applying the expectation gives (3.13).
The rest of the proof is showing that (3.12) holds. Fix (for now) some δ > 0, and let
−1 < w < −1+δ. In the t coordinates, suppose W (0) = w = 2t−1, so that −1 < w < −1+δ
corresponds to 0 < t < δ/2. Fix any t∗ ∈ (0, δ/2), and let w = 2t∗ − 1 be the starting point
ˆ t )t>0 solving (3.14), where W
ˆ has an infinite lifetime. Define a stopping
of the process (W

83

ˆ t = −1 + δ}. Then equation (3.8) implies
time by Tδ = inf{t > 0 : W

τt∗ =

∞
∞
4
4
1
ˆ 2 )dt ≥ 1
ˆ 2 )dt
e− κ t (1 − W
e− κ t (1 − W
t
t
4κ 0
4κ T
δ
4

= e− κ Tδ

∞
4
1
ˆ 2 )dt.
e− κ t (1 − W
t+Tδ
4κ 0

The right hand side is the stopping time for the process started at t = δ/2, so conditioned
4
on {Tδ = s}, we have τt∗ ≥ e− κ s τδ/2 in distribution. Therefore, Et∗ [1/τt∗ |Tδ = s] ≤
4

e κ s Eδ/2 1/τδ/2 . It follows that

Et∗

1
=
τt∗

∞

1
|T ∈ [n − 1, n) Pt∗ [n − 1 ≤ Tδ < n]
τt∗ δ

Et∗
n=1
∞

1

4

e κ n+1 Eδ/2

≤
n=0

τδ/2

Pt∗ [Tδ ≥ n].

We claim that Pt∗ [Tδ ≥ n] decays exponentially in n. Observe that

ˆ t < −1 + δ}, . . . , {
Pt∗ [Tδ ≥ n] = Pt∗ [{ sup W
0≤t≤1

sup

n−1≤t≤n

ˆ t < −1 + δ}].
W

ˆ is found, and it can be extended via the Kolmogorov
In [39], the transition density for W
consistency theorem to a process which can start at −1. Observe then that during each
time interval k − 1 ≤ t < k, the process lies above a coupling of the same diffusion started
ˆ started at −1, then
at −1. Therefore, if pδ = P0 [Tδ ≥ 1], where P0 is the law of W
ˆ t ≤ δ − 1] ≤ pδ . Moreover, by the Markov property, these couplings are
Pt∗ [supk−1≤t<k W
independent of the path before each time, so therefore Pt∗ [Tδ ≥ n] ≤ pn
δ.
4
1
1
≤ Eδ/2
e κ n pn
Thus, we have Et∗
n
δ . Now we need to show that δ can be
τt∗
τδ/2
84

chosen sufficiently small so that this infinite sum converges. What we need to know is if pδ
can be made sufficiently small as δ → 0. It is easier to see that this is the case if instead we
ˆ t ), which satisfies the SDE
look at θt = arccos(W

dθt = dBt +

4−κ
cot(θt )dt.
κ

(3.15)

Then θ is what is called a radial Bessel process. It is a diffusion with range in [0, π] represented
as a Brownian motion with strong drift directed towards the center of this interval. Also,
note that this drift is positive when θt is small (we are using that κ ≤ 4). Then

P0

as

sup θt ≤

0≤t≤1

≤ P[ sup Bt ≤ ] → 0
0≤t≤1

→ 0. It follows then that limδ→0 pδ = 0, and a fixed δ can be chosen so that the

expectation Et∗

1
τt∗

can be bounded uniformly in t∗ ∈ (0, δ/2). Speaking more precisely,

δ can be chosen small enough so that e4/κ pδ ≤ 1/2, and so for all t∗ ∈ (0, δ/2)
4

4

−n
e κ n Pt∗ [Tδ ≤ n] ≤ e κ n pn
δ ≤2 .

Therefore, to finish proving (3.12), it suffices to show that Eδ/2 1/τδ/2 < ∞.
Let θ0 = δ = arccos(−1 + δ), and define a stopping time by T = inf{t : θt ∈ {δ /2, π −
ˆ 2 = sin2 (θt ) imply that
δ /2}}. Then equation (3.8) and 1 − W
t
T

(4κ)τδ/2 ≥

T

4

e− κ s sin2 (θs )ds ≥ sin2 (δ /2)

0

0

85

4

4

e− κ s ds = Cδ (1 − e− κ T ).

Therefore,
1
Eδ/2
∞

= Cδ

1
Eδ/2

n=0

τδ/2

4
1 − e− κ T
∞

≤ Cδ
n=0

≤ Cδ Eδ/2

1
4

1 − e− κ T

|T ∈ [2−n−1 , 2−n ) Pδ/2 [T ∈ [2−n−1 , 2−n )]
1
4 −n−1
1 − e− κ 2

Pδ/2 [T ≤ 2−n ],

where Cδ is some constant which depends only on δ and κ. What we need is a good bound
on Pδ/2 [T ≤ ] as → 0. By the equation (3.15), whenever t ≤ T , we have

Bt − At ≤ θt − δ ≤ Bt + At,
4−κ
cot(δ /2) > 0.
κ
δ
δ
Let T = inf{t : θt = }, and T = inf{t : θt = π − }, so that T = min{T , T }. Then
2
2

where A =

Pδ/2 [T ≤ ] = Pδ/2 [T ≤ , T = T ] + Pδ/2 [T ≤ , T = T ]. Both of these probabilities will
be estimated similarly, so we will show that the first has sufficiently strong decay. Observe
that
Pδ/2 [T ≤ , T = T ] ≤ Pδ/2 [ inf θt ≤ δ /2, sup θt < π − δ /2]
0≤t≤

0≤t≤

= Pδ/2 [ inf θt − δ ≤ −δ /2, sup θt ≤ π − δ /2]
0≤t≤

0≤t≤

≤ P[ inf Bt − At ≤ −δ /2] = P[ sup Bt + At ≥ δ /2] ≤ P[ sup Bt ≥ δ /2 − A ],
0≤t≤

0≤t≤

0≤t≤

where P is the law of standard Brownian motion. Note that with δ fixed,

can be chosen

small enough so that this probability is nontrivial. Moreover, by the scaling property of

86

Brownian motion and the reflection principle, we get

=P

sup Bt ≥

δ /2 − A
√

0≤t≤1

Using the elementary bound P[N (0, 1) ≥ M ] ≤

Pδ/2 [T ≤ , T = T ] ≤ C

√

δ /2 − A
√

= 2P B1 ≥

.

C −M 2 /2
e
, this implies that
M

exp{−

√
((δ /2) − A )2
} ≤ C e−K/ ,
2

where C, K are both constants depending only on δ and κ. An identical bound can be found
for the event {T ≤ },so the same bound holds for Pδ/2 [T ≤ ].
Combining the inequalities of the previous paragraphs, we can conclude that

1
Eδ/2

τδ/2

∞

≤C
n=0

n

1
4 −n−1
1 − e− κ 2

n

2− 2 e−K2

.

n

The terms

e−K2

4 −n−1
1 − e− κ 2

are bounded, and therefore

1
≤C
Eδ/2
τδ

∞

n

2− 2 < ∞.

n=0

We will now make use of the results in Section 3.2, which originate from [37]. For
∗
∗
any N > 0 and x, y > 0, define the measure Px,y;N
by Px,y;N
[E] = P∗x,y [E\ΣN ], where

ΣN = {f ∈ Σ : Tf > N }. Note that this is not a probability measure. By Proposition 4, we

87

know that the local Radon Nikodym derivative P∗x,y with respect to Pκ is given by
dP∗x,y |Ft ∩Σt
dPκ |Ft ∩Σt

= I{τx,y >t} Mt (x, y) = I{τx,y >t} G(Xt (x), Yt (y))ft (x)ft (−y),

4 4

(3.16)

8

where G(x, y) = G−4,−4 (x, y) = x κ y κ (x + y)− κ is the (−4, −4)-backward SLE Green’s
function in the capacity parametrization. Let E ∈ Ft ∩ Σt . If t ≥ N , then P∗x,y;N [E] = 0. If
t < N , then

P∗x,y;N [E] := P∗x,y [E] − P∗x,y [E ∩ ΣN ] =

=

MN (x, y)
Mt (x, y)
dPκ −
dPκ
E∩ΣN G(x, y)
E G(x, y)

1
Mt (x, y) − Eκ [MN (x, y)|Ft ]dPκ .
G(x, y) E

Therefore, we conclude that
dP∗x,y;N |Ft ∩Σt
dPκ |Ft ∩Σt

=

I{t<N }
G(x, y)

(Mt (x, y) − Eκ [MN (x, y)|Ft ]) .

(3.17)

For each t ≥ 0, define a function Gt (x, y) by
Gt (x, y)
M (x, y) − Eκ [Mt (x, y)]
= 0
= P∗x,y [τx,y ≤ t].
G(x, y)
G(x, y)

The second equality follows by Proposition 4. By equation (3.10), and the fact that G(x, y)
is homogeneous of degree 0, it follows that Gt (x, y) = G1

x y
√ , √ . Also, G0 (x, y) = 0 for
t t

all x, y > 0, and limt→∞ Gt (x, y) = G(x, y)P∗x,y [τx,y < ∞] = G(x, y).
Let Cκ,t := 0∞ 0∞ Gt (x, y)dxdy = 0∞ 0∞ G(x, y)P∗ (τx,y ≤ t)dxdy, and let Cκ = Cκ,1 .
We claim that Cκ,t = tCκ , where Cκ ∈ (0, ∞). The equality follows from scale invarience

88

and the change of variables formula. In particular,
∞

Cκ,t =

0

∞

∞

Gt (x, y)dxdy =

0

∞

√
= ( t)2
0

0

∞
0

G1

x y
√ ,√
t t

dxdy

∞
0

G1 (x, y)dxdy.

Since Cκ = 0∞ 0∞ G(x, y)P∗x,y [τx,y < 1]dxdy, Lemma 16 and the fact that G(x, y) > 0
implies that Cκ > 0. Moreover, G is bounded, which combined with Lemma 16 implies that
Cκ < ∞.
Recall that Q1 = [0, ∞) × [0, ∞) denotes the first quadrant. Given a λ ∈ ΣW , we denote
the welding curve generated by λ by Φ : [0, ∞) → Q1 , where Φ(t) = (x, y) if x and −y are
absorbed at time t. That is, φ(x) = y and τx = τ−y = t. For U ⊂ Q1 measurable, we define
an increasing process (ΘU
t ) by

−1
ΘU
t = Cκ m+ {Φ (U ) ∩ [0, t]},

(3.18)

where m+ {·} is one dimensional Lebesgue measure on [0, ∞). Then ΘU
t is the capacity time
spent by Φ in the set U before t. Note that then dΘU
· is a kernel from (Σ, F) to [0, ∞)
and can be thought of as an F-measurable random measure on the line. Also, recall that
(x, y) → Px,y is a probability kernel from Q1 to Σ, and we can define a measure on Σ for
−4,−4

U ⊂ Q1 measurable by PU = PU

= U Px,y G(x, y)dxdy.

Theorem 6. Fix κ ∈ (0, 4). For any measurable U ⊂ Q1 ,

−4,−4 ˆ

PU

⊕Pκ = Pκ ⊗ dΘU
· ,

89

(3.19)

−4,−4

W∗ (Px,y
−4,−4

where MU = MU

←
−
−4,−4
⊕ Pκ ) ⊗ IU G−4,−4 (x, y)dxdy = W∗ (Pκ ) ⊗ MU
,

(3.20)

is the kernel from the set of welding curves W(ΣW ) to Q1 defined by

U
MU (Φ, E) = Φ∗ (dΘU
· ){E} = dΘ{Φ−1 (E)} .

To ensure clarity, we recall that the measures in equation (3.19) are measures on the
space Σ⊕ × Σ⊕ , and the measures in equation (3.20) are measures on ΣC × Q1 supported
on W(ΣW ) × Q1 . Throughout the rest of the chapter, for any measure µ on Σ, we will use
µ|t to denote the restricted measure µ|Ft ∩Σt . The proof of the theorem follows the same
strategy as the proofs of Theorems 4.1, 5.1, and 6.1 in [37].
Proof. For any t ≤ N , the renewal property of the backward Loewner equation and the
Markov property of

√

κBt imply that Mt (x, y)−Eκ [MN (x, y)|Ft ] = I{τx,y >t} |ft (x)||ft (−y)|GN −t (Xt , Yt ),

where Xt (x) = ft (x) − λt and Yt (y) = λt − ft (−y). Also, recall the welding curve satisfies
Φ(t) = (bt , at ), where bt = sup{x > 0 : τx ≤ t} and at = sup{y > 0 : τ−y ≤ t}. Therefore,
∞
0

∞

∞
0

Mt (x, y)−Eκ [MN (x, y)|Ft ]dxdy =
∞

bt

GN −t (Xt (x), Yt (y))|ft (x)||ft (−y)|dxdy

∞

=
0

at

∞

0

GN −t (x, y)dxdy = Cκ,N −t = (N − t)Cκ .

(3.21)

The second equality follows from a direct application of the change of variables formula,
since ft (−at ) = ft (bt ) = 0.
Let ξ be the measure on Q1 given by dξ = G(x, y)dxdy, and observe that ν((x, y), E) =
Px,y;N (E) is a kernel from Q1 to Σ. By Proposition 2, ξ · ν = Q Px,y;N G(x, y)dxdy
1
is absolutely continuous with respect to Pκ , and the local Radon-Nikodym derivatives are

90

given by
∞ ∞ dP
dξ · ν|t
x,y;N |t
=
G(x, y)dxdy
dPκ |t
dPκ |t
0
0
∞

∞

=
0

0

Mt (x, y) − Eκ [MN (x, y)|Ft ]
G(x, y)dxdy = Cκ (N − t) ∨ 0.
G(x, y)

The second equality comes from (3.17), and the last inequality comes from (3.21). Applying
Proposition 3, we get
∞
0

∞

Px,y;N G(x, y)dxdy = KCκ d(t∧N ) (Pκ ).

0

Taking N → ∞, it follows that
∞

PQ1 =

0

∞
0

Px,y G(x, y)dxdy = KCκ dt (Pκ ).

ˆ κ = PQ ⊗P
ˆ κ , and so Proposition 5 implies that
Therefore, KCκ dt (Pκ )⊗P
1

ˆ κ = Pκ ⊗ Cκ m+ .
PQ1 ⊗P

(3.22)

Note that this is (3.19) in the case U = Q1 .
We claim that for any (x, y) ∈ Q1 ,

ˆ ∗ (Px,y ⊕P
ˆ κ ) = W∗ (Px,y ⊕ Pκ ) ⊗ δ(x,y) ,
W

(3.23)

where δ(x,y) is the point mass measure on Q1 . Observe that Px,y is supported on {f ∈
ˆ ∗ (Px,y ⊕P
ˆ κ ) is the pushforward of the measure
ΣW : Wf (Tf ) = (x, y)}, and recall that W
ˆ ◦⊕
ˆ ⊕(f,
ˆ : Σ⊕ × Σ⊕ → ΣC × Q1 given by W(
ˆ g)) =
Px,y ⊗ Pκ under the transformation W

91

ˆ ⊕ g, Tf ) = (Φ, Φ(Tf )), where Φ is the welding induced by f ⊕ g. Thus, if E ⊂ ΣC and
W(f
U ⊂ Q1 ,

ˆ ∗ (Px,y ⊕P
ˆ κ ){E × U } = Px,y ⊗ Pκ {(f, g) : Φ = W(f ⊕ g) ∈ E and Φ(Tf ) = (x, y) ∈ U },
W

which proves the claim.
Integrating (3.23) over all (x, y) ∈ Q1 with respect to the measure G(x, y)dxdy gives

−
ˆ ∗ (Pκ ⊗ Cκ m+ ) = W
ˆ ∗ (PQ ⊕P
ˆ κ ) = W∗ (Px,y ⊕ Pκ )←
W
⊗ G(x, y)dxdy,
1

where the first equality comes from (3.22). It will follow that

←
−
W∗ (Px,y ⊕ Pκ ) ⊗ G(x, y)dxdy = W∗ (Pκ ) ⊗ M,

(3.24)

where M = MQ1 is the kernel from W(ΣW ) to Q1 given by M(Φ, U ) = m+ {Φ−1 (U )}.
ˆ : ΣW × [0, ∞) → ΣC × Q1 is injective by Lemma 12. For
To see this, observe that W
E ∈ W(ΣW ) and U ⊂ Q1 , let (Φ, z) ∈ E × U , then there is a unique λ ∈ ΣW and t ∈ [0, ∞)
with W(λ) = Φ and Φ(t) = z. Then

ˆ ∗ (Pκ ⊗Cκ m+ ){E×U } = Pκ ⊗Cκ m+ {W
ˆ −1 (E×U )} =
W

W −1 (E)

m+ {(W(λ))−1 (U )}dPκ (λ)

m+ {Φ−1 (U )}dW∗ (Pκ )(Φ) = W∗ (Pκ ) ⊗ M{E × U }.

=
E

Since the measures agree on all cylinder sets, they must be the same.
For any U ⊂ Q1 , the kernel MU is the restriction of M to W(ΣW ) × U , and so (3.20)

92

follows from restricting both sides of (3.24) to W(ΣW ) × U . Finally, if we apply the inverse
ˆ −1 to (3.20), we get equation (3.19).
map W

Corollary 1. Suppose U ⊂ [0, ∞) × [0, ∞) with

U

G−4,−4 (x, y)dxdy < ∞. If the laws

of the welding curve of an extended BSLEκ (−4, −4) process started from (0; x, −y) are integrated against IU G−4,−4 (x, y)dxdy, it results in a bounded measure on curves which is
absolutely continuous with respect to the law of the BSLEκ welding curve, and the RadonNikodym derivative is |MU (Φ, Q1 )| = Cκ m+ (Φ−1 (U )).
Proof. For any measurable E ⊂ W(ΣW ), (3.20) implies that

U

←
−
W∗ (Px,y ⊕ Pκ ){E}G(x, y)dxdy = W∗ (Px,y ⊕ Pκ ) ⊗ IU G(x, y)dxdy{E × Q1 }

= W∗ (Pκ ) ⊗ MU {E × Q1 } =
=
E

3.6

E

MU (Φ, Q1 )dW∗ (Pκ ){Φ}

Cκ,1 m+ {Φ−1 (U )}dW∗ (Pκ ){Φ}.

One-point estimate

The goal of this section is to establish a one-point estimate for the BSLE welding, similar to
the one-point estimate used to establish the Green’s function for forward SLE. After doing
this, we will see that G(x, y) = G−κ−4,−κ−4 (x, y) = x1+4/κ y 1+4/κ (x + y)−4−κ/2−8/κ serves
as an appropriate Green’s function to use to condition welding x and y together. This is the
only special case in which we have been able to construct Ga,b via a geometrically motivated
93

limit procedure.
Fix any x, y > 0. We will again be doing Ito’s formula calculations using the definitions
√
from Section 3.3. For clarity, we will review the definitions. Let Xt = ft (x) − κBt and
√
Yt = κBt − ft (−y) so that both X and Y are positive. Ito’s formula implies that
√
√
2
2
dXt = − κdBt −
dt, and dYt = κdBt − dt.
Xt
Yt
Moreover,
−2
d(Xt + Yt )
=
dt.
Xt + Yt
Xt Yt

(3.25)

X −Y

Define Wt = Xt +Yt as before, which is a process which stays in (−1, 1) for as long as it
t t
exists. It only fails to exist if τx = τ−y , where τz is the time so that z is swallowed by the
BSLE traces. If this does not happen (which is Pκ - a.s.), then Wt is constant at ±1 after
τ = τx,y := min{τx , τ−y }. What we want to estimate is P(E ), where E = {Xτ + Yτ ≤ }.
Theorem 7. For any x, y > 0, with the notation above, there is a constant C < ∞ depending
only on κ so that

1
lim − 2 (κ+4) P(E ) = Cx1+4/κ y 1+4/κ (x + y)−4−κ/2−8/κ := G(x, y).
→0

Proof. By Ito’s formula,

(3.26)

√
−2 κ
4Wt
dWt =
dBt +
dt.
Xt + Yt
Xt Yt

Performing the random time change u(t) = 0t

κ
dr, we get
X r Yr

˜t = − 1 − W
˜ 2 dB
˜t + 4 W
˜ t dt,
dW
t
κ

94

(3.27)

˜ t = W −1 . Recall that W
ˆ t is a radial Bessel process.
where B˜s is a Brownian motion and W
u (t)
Observe that uτ is the lifetime of a process satisfying (3.7). By [39], there exists a constant
C depending only on κ so that for any t > 0,

P{uτ > t} ≤ C exp{

By equation (3.5), we have Xt + Yt =

8
−1
(2 + )t}.
2
κ

(3.28)

κ
if and only if ut = − ln
2

x+y

:= h( ), and

E = {uτ > h( )}. Thus, equation (3.28) implies

P(E ) = P{uτ >

−κ
−1
ln
≤ C exp{ (2 + 8/κ)
2
2

−κ
ln
2

x+y

x+y

}=C

}

1 (κ+4)
2

x+y

.

(3.29)

In order to find the exact value of equation (3.26), we make use of a more precise bound
for P(uτ > t). Equation (B.14) in [39] states that

|P{T > t} − 2˜
p(t, w)| ≤ Ce−(6+12/κ)t ,

(3.30)

where w = (x − y)/(x + y) and

p˜(t, w) = Cκ (1 − w2 )1+4/κ e−(1+4/κ)t .

Inequality (3.29) implies that
3κ+6

|P(E ) − 2˜
p(h( ), w)| ≤ C exp{−(6 + 12/κ)(−κ/2) ln

95

x+y

}=C

x+y

.

Multiplying through by −1/2(κ+4) gives

| −1/2(κ+4) P(E ) − 2 −1/2(κ+4) p˜(h( ), w)| ≤ C

3κ+6
1
5/2κ+4 → 0
x+y

as → 0. Thus,
lim −1/2(κ+4) P(E ) = lim 2 −1/2(κ+4) p˜(h( ), w),
→0

→0

if this second limit exists.
By the definition of p˜, we have

p˜(h( ), w) = Cκ

(1 − w2 )1+4/κ

1/2(κ+4)

x+y

,

and therefore, using w = (x − y)/(x + y), we get

−1/2(κ+4) p˜(h(

), w) = Cκ

(1 − w2 )1+4/κ
(x + y)1/2(κ+4)

for every > 0.

96

= Cκ x1+4/κ y 1+4/κ (x + y)−4−κ/2−8/κ

Chapter 4
Measure driven chordal Loewner
evolution

4.1

Theorem statement and definitions

For any T > 0, let NT denote the set of positive measures µ on [0, T ] × R whose first
coordinate marginal is Lebesgue measure m on the Borel subsets of [0, T ]. That is, for any
Borel set E ⊂ [0, T ], we have µ(E × R) = m(E). We also assume that the support of µ is
compact. Define RT to be the minimal R > 0 so that µ|[0,T ]×R = µ|[0,T ]×[−R,R] . This set of
measures has a natural topology induced by weak convergence of measures. That is, µn → µ
in NT if for every bounded φ ∈ C([0, T ] × R), where C(X) = continuous functions on X, we
have
lim

n→∞ [0,T ]×R

φdµn →

φdµ.
[0,T ]×R

Similarly, define the space N by µ ∈ N if µ is a measure on [0, ∞) × R which has
locally compact support and whose first coordinate marginal is Lebesgue measure. Then we
define the natural projections PT : N → NT by PT (µ) = µ|[0,T ]×R . The topology on N is
determined by µn → µ in N if PT (µn ) → PT (µ) for every positive T .
Let G be the set of familes of conformal maps (gt ) with gt : H\Kt → H, where (Kt )
is a family of growing H-hulls with hcap(Kt ) = 2t for every t. We will always use (ft ) to
97

represent the family of inverse maps ft : H → H\Kt . We put a topology on G induced by
(gtn ) → (gt ) if ftn (z) → ft (z) locally uniformly in (t, z).
For µ ∈ N , we say that (gt ) solves the chordal Loewner equation driven by µ if

gt (z) − z =

Φ(u, gs (z))dµ(s, u),
[0,t]×R

Φ(u, z) =

2
,
z−u

(4.1)

for every z ∈ H\Kt , where Kt is the set where the solution blows up by time t.
Theorem 8.

a) There is a one-to-one correspondence between measures µ ∈ N and con-

tinuously growing families of H-hulls (Kt ) with hcap(Kt ) = 2t. This correspondence is
given by the unique solution (gt ) of (4.1) driven by µ, where gt = gKt .
b) Suppose {µn } and µ in N have solutions to (4.1) given by (gtn ) and (gt ) respectively.
Assume further that for all t > 0, the associated hulls {Ktn }n∈N are uniformly bounded.
Then µn → µ in N if and only if the associated hulls Ktn → Kt in the Carath´eodory
topology.
The basic argument for the proof is as follows. First, given a measure µ ∈ N , we use
the general form of the chordal Loewner equation in [7] to approximate a disintegration
n
of µ as dµn
t dt → dµt dt = dµ. For each n, the family of measures dµt define a family of

chordal Loewner hulls (Ktn )t>0 with corresponding Loewner maps (gtn )t>0 . We show that
subsequences of these converge to a family of conformal maps (gt ) and chordal hulls (Kt )
such that gt solves (4.1). To show the correspondence in the other direction, we start with
a family of hulls (Kt ). For any hull K with hcap(K) = 2T we show that there is some
measure µT so that µT can be associated with K. For the whole process (Kt ), we use this
construction to build measures µδn which account for growth in each interval of length δ.
98

Taking the limit of this approximation as δ → 0, we get a measure µ which corresponds to
the family of hulls (Kt ).

4.2

Preliminary results

Theorem 8 mentions convergence in the Carath´eodory sense. The definition of this convergence can be found in [26], and will be restated here. Let {Dn }n∈N and D be domains in
C. We say that Dn → D in the Carath´eodory sense if
(i) For any compact K ⊂ D, there is an N so that K ⊂ Dn for all n ≥ N .
(ii) For any z ∈ ∂D, there exists a sequence zn ∈ ∂Dn so that zn → z.
Hulls K n are said to converge in the Carath´eodory sense to K if their complements converge.
That is, if H\K n → H\K in the Carath´eodory sense. The following lemma can be found in
[26].
Lemma 17. Suppose that Dn → D as domains, gn : Dn → C are conformal, and g : D → C
is a function so that gn → g locally uniformly on D. Then either g is constant or g is
conformal. In the latter case, the following statements hold:
(a) gn (Dn ) → g(D) in the Carath´eodory topology,
(b) gn−1 → g −1 locally uniformly in g(D).
Here we provide the proof that the assumptions of the classical existence and uniqueness
theorem are satisfied for the chordal Loewner ODE driven by a weakly continuous family of
measures on R as introduced in [7].

99

Proposition 7. Suppose that {νt }t≤T is a family of positive Borel measures on R so that
t → νt is continuous in the weak topology. Further, assume that for all t, there exists some
Mt > 0 so that νs (R) ≤ Mt and supp{νs } ⊂ [−Mt , Mt ] for all s ≤ t. Then for all z ∈ H,
there exists a unique solution to the differential equation

∂t gt (z) =

2
dνt (z),
R gt (z) − u

g0 (z) = z

on some maximal interval [0, Tz ).
Proof. Fix z ∈ H and T > 0. Define R = {(t, w) : 0 ≤ t ≤ T, Im(w) ≥ Im(z)/2}. Let

ψ(t, w) =

2
dνt (u),
R w−u

so that ∂t gt (z) = ψ(t, gt (z)). By the classical local and maximal existence and uniqueness
theorems for ODEs, if both ψ and ∂w ψ are continuous in some [0, t] × B(z, r) ⊂ R, then
there exists a solution to the ODE in some maximal time interval [0, Tz ).
First, we compute ∂w ψ(t, w). Observe that if w, ξ have imaginary parts at least Im(z)/2,
then
ψ(t, w) − ψ(t, ξ)
1
=
w−ξ
R (w − ξ)
=

2
2
−
w−u ξ−u

dνt (u)

2
ξ−w
−2
·
dνt (u) →
dν (u),
2 t
R w − ξ (w − u)(ξ − u)
R (w − u)

where the limit is as ξ → w. This follows by the dominated convergence theorem, since
|ξ − u|, |w − u| ≥Im(z)/2 and νt is a finite measure.
Next, we will prove that ψ is continuous in the desired region. Suppose that (tn , wn ) →
(t, w), with (tn , wn ) ∈ R. Then

100

|ψ(tn , wn ) − ψ(t, w)|
≤

2
2
dνtn (u) −
dνtn (u) +
R wn − u
R w−u

Since Im(w) ≥Im(z)/2, the function u →

2
2
dνtn (u) −
dνt (u) .
R w−u
R w−u

2
is uniformly bounded, so weak continuity
w−u

implies that the second term approaches 0. To estimate the first term, the map w →
(w − u)−1 is Lipschitz with constant C depending only on the fixed z, so

2
2
dνtn (u) −
dνtn (u) ≤ C
|wn − w|dνtn (u) ≤ CMT |wn − w| → 0.
R wn − u
R w−u
R
Thus, ψ is continuous in the desired region. The proof for ∂w ψ will be identical based on
the above formula, only it will use the Lipschitz constant for w → (w − u)−2 .

We need to establish an important result about H-hulls. In [36], it is proven that there
∗ so that µ
is a positive measure µK with support in SK
K =hcap(K), and

fK (z) − z =

−1
dµ (u).
∗ z−u K
SK

Lemma 18. If K is an H-hull, then for all z ∈ H,

|fK (z) − z| ≤ 7diam(K).

Proof. Suppose without loss of generality that K ⊂ Kr := {z ∈ H : |z| ≤ r}, where
∗ ⊂ S ∗ = [−2r, 2r].
r = max{|z| : z ∈ K}. In [7], it is proven that hcap(Kr ) = r2 and SK
Kr

101

Fix any L > 2r, and suppose that |z| ≥ L. Then

|fK (z) − z| ≤

µK
r2
1
dµK (u) ≤
≤
.
∗ z−u
L − 2r
L − 2r
SK

Suppose that |z| ≤ L. Then the maximum modulus principle implies that

|fK (z)| ≤

sup
{|w|=L}∪{−L<w<L}

|fK (w)|.

Since fK can be extended to an analytic function on C\SK (see [26]), the maximum in
{|w| ≤ L} occurs either on AL := {w ∈ H : |w| = L} or on SK . Extending fK continuously
to H, we get that |fK (z)| ≤diam(K) ≤ r as z → SK . For z ∈ AL , we’ve already shown that
r2
|fK (z)| ≤ |z| +
= L. Thus, picking L = 3r, we get that
L − 2r

|fK (z) − z| ≤ |z| + max{r, |z| +

4.3

r2
} ≤ 3r + max{r, 3r + r} = 7r.
3r − 2r

A disintegration lemma

We begin by demonstrating how we disintegrate a measure µ ∈ N .
Lemma 19. Let µ ∈ N . Then there exists a family of measures (µn
t )t>0,n∈N on R so that
(a) Each µn
t is a probability measure on R.
(b) For every n ∈ N, the map t → µn
t is continuous with respect to the weak topology on
measures on R.

102

n
(c) The measure given by dµn
t dt is in N , and dµt dt → µ as n → ∞ in N .

(d) For almost every t ∈ [0, ∞), there is a probability measure µt so that µn
t → µt weakly.
Also, dµt dt = dµ.
Proof. First, we must define the measures µn
t . For each n ∈ N, t > 0, define a linear
functional Ln,t : Cc (R) → C by

Ln,t (φ) = n

φ(u)dµ(s, u).

(4.2)

[t,t+1/n]×R

To see that this map is well defined, observe that

|Ln,t (φ)| ≤ n

[t,t+1/n]×R

|φ(u)|dµ(s, u) ≤ φ ∞ nµ([t, t + 1/n] × R) = φ ∞ < ∞.

The map Ln,t is clearly a positive linear functional. By the Riesz representation theorem,
there exists a Radon measure µn
t defined on the Borel sets of R so that equation (4.2)=
n
R φdµt .

To prove (a), for every k ∈ N, define φk ∈ Cc (R) so that φk (u) = 1 for u ∈ [−k, k],
and φk (u) → 1 monotonically for every u. By the monotone convergence theorem (applied
twice) and equation (4.2),

µn
t (R) = lim

k→∞ R

φk dµn
t = lim n
k→∞

[t,t+1/n]×R

φk (u)dµ(s, u)

= nµ([t, t + 1/n] × R) = 1.
We will now make several arguments about the measures converging weakly on R. For
weak convergence, we need to test all bounded continuous functions on R. However, we
103

are testing the local properties of the measures µ before some time T . We assume that
our measure µ, when restricted to [0, T ] × R, has support contained in [0, T ] × [−RT , RT ].
Therefore, if φ is any bounded continuous function, there exists some φ∗ ∈ Cc (R) so that
φ∗ = φ on [−RT , RT ]. Hence, for all weak convergence arguments below, we will make this
reduction automatically and work only with continuous functions with compact support.
To see (b), suppose that tk → t in [0, ∞). Then tk → t in [0, T ] for some T . Let φ ∈ Cc (R)
be arbitrary. Then
φdµn
t =n
k

R

=n

φ(u)dµ(s, u)
[tk ,tk +1/n]×R

I[t ,t +1/n]×R (s, u)φ(u)dµ(s, u).
k k

[0,T ]×R

(4.3)

Then I[t ,t +1/n]×R (s, u)φ(u) → I{[t,t+1/n]×R} (s, u)φ(u) µ-almost everywhere, and is domik k
nated by φ ∞ ∈ L1 ([0, T ]×R, µ). Thus, Lebesgue’s dominated convergence theorem implies
that
lim (4.3) = n

k→∞

[0,T ]×R

φdµn
t.

I{[t,t+1/n]×R} (s, u)φ(u)dµ(s, u) =
R

Now we must show c), starting with dµn
t dt ∈ N . For any Borel set E ⊂ R, part (a)
implies that
E

1dµn
t dt =
R

dt = m(E).
E

Also, if t ≤ T , then it is clear that the support of µn
t is contained in [−RT +1 , RT +1 ].
Therefore, dµn
t dt ∈ N . To show convergence, suppose that φ ∈ Cc ([0, T ] × R). For any
T > 0,
[0,T ]×R

φ(t, u)dµn
t (u)dt =

[0,T ]

φ(t, u)dµn
t (u) dt
R

T

=

n
0

φ(t, u)dµ(s, u) dt
[t,t+1/n]×R

104

s

=

s

T

+
[0,1/n]×R 0

+
[1/n,T ]×R s−1/n

nφ(t, u)dtdµ(s, u).
[T,T +1/n]×R s−1/n

Since φ has compact support, fixing n guarantees that this function is L1 (µ × m), so the
above use of Fubini’s theorem is justified. We will show that the first and third terms go to
0, and the middle term converges to [0,T ]×R φ(s, u)dµ(s, u).
To bound the first term, observe that
s

s

nφ(t, u)dtdµ(s, u) ≤ φ n

1dtdµ(s, u)
[0,1/n]×R 0

[0,1/n]×R 0

sdµ(s, u) ≤ φ n

= φ n
[0,1/n]×R

1
φ
dµ(s, u) =
,
n
[0,1/n]×R n

which converges to 0 as n goes to infinity. The third integral converges to 0 by a similar
argument. Thus, it suffices to show that

lim

n→∞ [1/n,T ]×R

s
1
φ(t, u)dt dµ(s, u) =
φ(s, u)dµ(s, u).
1/n s−1/n
[0,T ]×R

(4.4)

For each u, the map t → φ(t, u) is continuous on [0, ∞), so the fundamental theorem of
calculus implies that
s
1
φ(t, u)dt = φ(s, u)
n→∞ 1/n s−1/n

lim

for each s. Thus, for µ-almost every (s, u),
s

I{[1/n,T ]×R} (s, u) n

φ(t, u)dt
s−1/n

105

→ I{[0,T ]×R} (s, u)φ(s, u)

as n → ∞. Also, for every n ∈ N, s > 0, and u ∈ R, we have
s

I{[1/n,T ]×R} (s, u)n

s−1/n

φ(t, u)dt ≤ φ ∞ I[0,T ]×[−N,N ] (s, u) ∈ L1 (µ),

where the support of φ is contained in [0, T ]×[−N, N ] for some N > 0. Therefore, Lebesgue’s
dominated convergence theorem implies that (4.4) holds, which completes the proof of (c).
We now prove (d). Since µn
t is a collection of Radon probability measures with uniformly
compact support for each t, there is a subsequential weak limit µt . Let {φk }k∈N be a dense
countable subset of Cc (R), which is separable. We claim that for each k, there exists a subset
Ek ⊂ [0, ∞) so that m(Ekc ) = 0, and for all t ∈ Ek we have

lim
φ (u)dµn
t (u)
n→∞ R k

=

φk (u)dµt (u).
R

To see this, define a measure νk on R by νk (E) := E×R φk (u)dµ(s, u), which is absolutely
continuous with respect to Lebesgue measure. By ([28], Theorem 7.14) we have for m-almost
all t (where a.e. depends on φk , hence the set Ek ),

φk dµn
t =
R

Thus, limn
φk dµn
t →

ν [t, t + 1/n]
dν
1
φk (u)dµ(s, u) = k
→ k (t).
1/n [t,t+1/n]
m[t, t + 1/n]
dm

φk dµn
t exists, and there is subsequential limit equal to

φk dµt , and therefore

φk dµt for each k.

Let E = ∩k Ek , so that m(E c ) = 0, and suppose t ∈ E. Let φ ∈ Cc (R). We claim that
φdµn
t →

φdµt as n → ∞. Let

> 0. Then there exists some k so that φk − φ ∞ < .

106

Then for sufficiently large n,

φ(u)dµn
t (u) −

φ − φk dµn
t +

≤
R

φ(u)dµt (u)
R

R

φk dµn
t −
R

φk − φdµt ≤ + + .

φk dµt +
R

R

Thus, µn
t → µt weakly for all t ∈ E, and hence for m-almost every t.
The final claim is true because for any φ ∈ Cc ([0, T ] × R), for any u and almost any s, we
have R φ(s, u)dµs (u) = limn→∞ R φ(s, u)dµn
s (u). Therefore, the dominated convergence
theorem implies that
t

t

φ(s, u)dµs (u)ds =
0

R
t

φ(s, u)dµn
s (u)ds =

= lim

n→∞ 0

lim

0 n→∞ R

R

φ(s, u)dµn
s (u)ds

φ(s, u)dµ(s, u).
[0,t]×R

The last equality follows from part (c), since dµn
t dt → dµ in N .

Remark: For each t > 0 so that µt as in part (d) exists, we can exactly repeat this
construction with the functional

φ→n

φ(u)dµ(s, u)
[t−1/n,t]×R

for each n. The resulting measures will converge to the same µt for almost all t. Therefore,

107

if h < 0, we can use the convention

φ(s, u)dµ(s, u) = −
[t,t+h]×R

φ(s, u)dµ(s, u)
[t+h,t]×R

as in Riemann integration to discuss differences gt+h (z) − gt (z) without making reference to
whether h is positive or negative.

4.4

Elementary calculations

Next, we will make several calculations assuming that (4.1) driven by µ has a solution (gt ).
These calculations will then be applied to the measures dµn = dµn
t dt, which have solutions
(gtn ) for each n by Proposition 7. Moreover, the maps gtn are all conformal, and a solution
to (4.1) will be constructed as a locally uniform limit of gtn , and so we can treat any solution
as conformal. We state and prove these lemmas in their more general state since we will
need them after a solution is proven to exist. First, we have to prove two Lipschitz estimates
which will be used repeatedly.
Lemma 20.

a) If z, w ∈ H with imaginary parts at least h > 0 and u ∈ R, then
2
|Φ(u, z) − Φ(u, w)| ≤ 2 |z − w|.
h

b) Suppose (gt ) is a solution to the chordal Loewner equation driven by µ ∈ N . Then for
any z ∈ H and t1 < t2 ≤ T < τz ,

|gt2 (z) − gt1 (z)| ≤

108

2
|t − t1 |.
Im(gT (z)) 2

Proof. To prove a), observe that |z − u| ≥ h and |w − u| ≥ h. Therefore,

z−w
2
2
2
−
=2
≤ 2 |z − w|.
z−u w−u
(z − u)(w − u)
h

|Φ(u, z) − Φ(u, w)| =

To prove b), observe that (4.1) implies that the imaginary part of gt (z) is a decreasing
function of t. Therefore,

|gt2 (z) − gt1 (z)| =

≤

2
2
dµ(s, u) ≤
dµ(s, u)
[t1 ,t2 ]×R gs (z) − u
[t1 ,t2 ]×R |gs (z) − u|

2
2
2
dµ(s, u) ≤
dµ(s, u) =
|t − t1 |.
Im(gT (z)) 2
[t1 ,t2 ]×R Im(gs (z))
[t1 ,t2 ]×R Im(gT (z))

Lemma 21. If (gt ) is a solution to the chordal Loewner equation driven by µ ∈ N and
z ∈ H. Then for almost any t < τz ,

∂t gt (z) =

2
dµt (u),
R gt (z) − u

where µt is the measure from Lemma 19 part d).
Proof. This follows from Lemma 19 part d), which says that dµt (u)dt = dµ(t, u). In particular, if T < τz , Lemma 19 part d) implies that for t < T ,
t

gt (z) − z =

Φ(u, gs (z))dµ(s, u) =
[0,t]×R

0

2
dµs (u)ds.
R gs (z) − u

Since
2
2
dµs (u) ≤
∈ L1 ([0, T ]),
g
(z)
−
u
Im(g
(z))
R s
T
109

2
the fundamental theorem of calculus implies that ∂t gt (z) = R g (z)−z
dµt (z) for almost every
t

t.
We can now prove uniqueness.
Proposition 8. A solution of the chordal Loewner equation driven by µ ∈ N must be unique.
Proof. Fix z ∈ H, and suppose that gt and g˜t are two solutions of the integral equation

gt (z) = z +

2
dµ(s, u)
[0,t]×R gs (z) − u

up to some time T + . Then {Im(gt )}t≤T is decreasing and bounded below by Im(gT ). The
same is true for g˜t , so we can assume that there is a constant C so that for all t ≤ T , we have
Im(gt ) and Im(˜
gt ) are bounded below by C. By Lemma 21, for almost all t ≤ T (depending
on z, which is fixed), we know that

∂t (gt − g˜t ) = 2

g˜t − gt
dµt (u).
gt − u)
R (gt − u)(˜

Therefore,

|∂t (gt − g˜t )| ≤ 2

|˜
gt − gt |
2
2
dµt ≤ 2
|˜
gt − gt |dµt (u) = 2 |gt − g˜t |.
gt − u)|
C R
C
R |(gt − u)(˜

Since gt and g˜t are absolutely continuous by Lemma 20 , we get that
t

|gt − g˜t | ≤

|∂s (gs − g˜s )|ds ≤
0

t
2
|gs − g˜s |ds.
C2 0

By Gronwall’s inequality applied to |gt − g˜t |, it follows that gt = g˜t for all t ≤ T . Since T
was an arbitrary number below the lower blow-up time, it follows that gt = g˜t for all t.
110

Next, we need to derive some basic continuity and differentiability properties of the
inverse function f = ft (z) = gt−1 (z).
Lemma 22. Suppose that g(t, z) = gt (z) solves the chordal Loewner equation driven by
µ ∈ N , and let ft (z) = gt (z)−1 . Then both f and f are continuous in the variables (t, z).
Proof. The main part of this lemma is that if (Kt ) is an increasing family of hulls with
hcap(Kt ) = 2t, then the map t → H\Kt is continuous in the Carath´eodory topology. Then
Lemma 17, part (b) implies that if tn → t, then ftn → ft locally uniformly in H\Kt , and
therefore ftn → ft locally uniformly in H\Kt .
It suffices to prove that H\Ktn → H\Kt in the cases where tn increases to t and tn
decreases to t separately. First, assume that tn decreases to t. Then Kt ⊂ Ktn for all n. I
claim that ∩n Ktn = Kt . Let K = ∩n Ktn , which is an H-hull with Kt ⊂ K , and assume that
K = Kt . Then hcap(K ) >hcap(Kt ). However, 2tn =hcap(Kn ) ≥hcap(K ) >hcap(Kt ) =
2t, which contradicts the assumption that tn → t. Thus, Kt = ∩n Ktn . The properties for
Carath´eodory convergence are readily verified
Now, suppose that tn increases to t. Note that we do not necessarily have ∪n Ktn = Kt ,
so we must argue differently. Observe that H\Kt ⊂ H\Ktn for each n, so gtn is defined
on H\Ktn for each n. Let Dn = D = H\Kt . Then Dn → D clearly. Also, gtn → gt
locally uniformly, and gt is not constant. By Lemma 17 (a), we have that gtn (H\Kt ) → H.
Applying it again to the inverse maps yields H\Ktn → H\Kt .

n
Lemma 23. Let µ ∈ N , and let dµn = dµn
t (u)dt, where µt is as in Lemma 19. Then there

111

are functions (gtn ) which solve (4.1) driven by µn . If ftn = (gtn )−1 , then

ftn (z) − z = −

2(fsn ) (z) n
dµ (s, u)
[0,t]×R z − u

(4.5)

for each z ∈ H, t > 0, n ∈ N.
Proof. For each n, Lemma 19 part b) and Proposition 7 imply that there is a solution of

∂t gtn (z) =

2
dµn
t (z),
n
g
(z)
−
u
R t

g0n (z) = z,

which is the solution of (4.1) driven by µn . This equation holds for all t < τz . If ftn = (gtn )−1 ,
the chain rule then implies that

∂t ftn (z) =

−2(ftn ) (z) n
dµt (u),
z−u
R

f0n (z) = z.

Integrating this gives with respect to t yields (4.5).

We are now almost ready to prove existence of a solution to the chordal measure driven
Loewner equation. In order to use a limiting argument, we need some sort of control on the
size of the hulls {Ktn }t≤t0 for any fixed t0 . This control will be provided by the following
two lemmas.
Lemma 24. Suppose (gt ) is a solution of the chordal Loewner equation driven by µ ∈ N .
√
For each t, define Mt = max{ t, Rt }. Then for all z ∈ Kt , we have |z| ≤ 4Mt . Moreover,
if |z| > 4Mt , then |gs (z) − z| ≤ Mt for all 0 ≤ s ≤ t.
Remark: This lemma is an extension of Lemma 4.13 in [7], where it is proven for the
112

chordal Loewner equation driven by a continuous function. The proof is different, however.
The argument follows the standard proof for the existence of a solution of an ODE.
Proof. Fix t > 0 and z ∈ H with |z| ≥ 4Mt . Recall that Rt is defined to be Rt = inf{R >
0 : µ|[0,t]×R is supported in [0, t] × [−R, R]}. We will use a Picard iteration argument to
show that τz ≥ t, where τz is the lifetime of (4.1) at z. Let w ∈ B(z, Mt ), which implies
|w| ≥ 3Mt . For any u ∈ [−Rt , Rt ], we have |w − u| ≥ |w| − |u| ≥ 3Mt − Mt = 2Mt .
Therefore, if Φ(u, z) =

2
, we have for any w, w ∈ B(z, Mt ),
z−u

|Φ(u, w) − Φ(u, w )| =

2|w − w |
1
2(w − w )
≤
=
|w − w |.
(w − u)(w − u)
4Mt2
2Mt2

This implies that Φ(u, w) has Lipschitz constant

1
for w ∈ B(z, Mt ). Also, if w ∈
2Mt2

B(z, Mt ) and u ∈ [−Rt , Rt ],
|Φ(u, w)| ≤

2
1
=
.
2Mt
Mt

Then the restriction of Φ to [−Rt , Rt ] × B(z, Mt ) satisfies Φ ∞ ≤ 1/Mt .
To perform the Picard iteration, fix an a > 0 to be determined later, and define an
operator Λ on the set of continuous functions from [0, a] to B(z, Mt ) by

Λ(φ)(r) = z +

Φ(u, φ(s))dµ(s, u),

0 ≤ r ≤ a.

[0,r]×R

We will show that for each a < t, Λ(φ) will be a continuous map from [0, a] to B(z, Mt ). If
a < t, then a < Mt2 by construction of Mt . For r ≤ a,

|Λ(φ)(r) − z| =

Φ(u, φ(s))dµ(s, u)
[0,r]×R

113

|Φ(u, φ(s))|dµ(s, u) ≤

≤
[0,r]×R

a
≤ Mt .
Mt

(4.6)

Therefore, Λ(φ) sends [0, a] into B(z, Mt ).
Next, we need to show that Λ is a contraction. For any continuous φ1 , φ2 : [0, a] →
B(z, Mt ), the Lipschitz estimate implies that for any r ∈ [0, a],

|Λ(φ1 )(r) − Λ(φ2 )(r)| ≤

≤

[0,r]×R

|Φ(u, φ1 (s)) − Φ(u, φ2 (s))| dµ(s, u)

1
a
φ1 − φ2 ∞ dµ(s, u) ≤ 2 φ1 − φ2 ∞ ,
2
Mt
[0,a]×R 2Mt

where a/Mt2 < 1. Thus, Λ is a contraction between Banach spaces, and the Banach fixed
point theorem implies that there is a unique continuous φ : [0, a] → B(z, Mt ) so that

φ(r) = Λ(φ)(r) = z +

2
dµ(s, u),
[0,r]×R φ(s) − u

0 ≤ r ≤ a.

By Proposition 8, φ(r) = gr (z) for all r ≤ a < t. Thus, τz ≥ t. Moreover, if |z| > 4Mt , this
construction can be stretched to a = t, and so applying (4.6) to gt (z) proves |gs (z) − z| ≤ Mt
for all s ≤ t.

n
Lemma 25. If µ ∈ N , let µn
t be as in Lemma 19. For each n, let (gt ) be the Loewner maps
n
driven by {µn
t }t≤T with corresponding H-hulls (Kt )t≤T . Then for every t0 > 0, there exists

some Mt0 > 0 so that Ktn ⊂ {z ∈ H : |z| ≤ Mt0 }.
0
Proof. By Lemma 24, it suffices to show that the support of the measures {µn
t }t≤t0 ,n∈N are
uniformly bounded. We have assumed that µ restricted to [0, t0 + 1] × R is supported in
[0, t0 + 1] × [−R, R] for some R > 0. Suppose that x ∈ R and > 0 so that (x − 2 , x + 2 ) ×
114

[t, t + 1/n) ⊂ ([0, t0 + 1] × [−R, R])c . In particular, µ((x − , x + ) × [t, t + 1/n]) = 0. Let
φk ∈ Cc (R) be a monotonic sequence converging to I(x− ,x+ ) . Then

µn
t (x − , x + ) = lim

k→∞ R

φk (u)dµn
t (u) = lim n
k→∞

[t,t+1/n]×R

φk (u)dµ(s, u)

= nµ((x − , x + ) × [t, t + 1/n]) = 0.
n
Since the supports of the measures µn
t are uniformly bounded, it follows that the hulls Kt

are bounded independently of n.

4.5

Proof of existence

Using the boundedness of hulls from Lemma 25, we can prove equicontinuity of some subsequence of any family f n of inverse functions associated with a sequence of measures µn ∈ N
associated with µ.
Lemma 26. Let µ ∈ N , and suppose (gtn )t≥0 is the family of conformal maps which solve
the chordal Loewner equations driven by the measures dµn (t, u) = dµn
t (u)dt for each n. Then
there exists a family of conformal maps (gt ) so that a subsequence of (gtn ) converges to (gt )
in G.
Remark: The proof of this lemma will hold for any µn ∈ N with µn → N as long as
(4.5) holds at each n, and if the hulls (Ktn ) are uniformly bounded for each t. We will do
this to prove that Theorem 8 part b) holds.
Proof. Fix T < ∞. Applying Lemma 18 to each member of the family (Ktn )t≤T,n∈N , and us115

ing the assumption that the hulls are uniformly bounded, we get that the family (ftn )t≤T,n∈N
is normal in H. We will use this normality and equation (4.5) to show that the functions
f n are equicontinuous in both t and z on any compact set [0, N ] × RN ⊂ [0, ∞) × H, where
RN = [−N, N ] × [1/N, N ]. Since (ftn )t≤T,n∈N is a normal family on H, so are the families
of derivatives (ftn )t≤N,n∈N and (ftn )t≤N,n∈N . Since RN ⊂ H is compact, this implies that
there are constants CN , CN < ∞ with |(fsn ) (z)| ≤ CN and |(fsn ) (z)| ≤ CN for all s ≤ N
and z ∈ RN . Therefore,

|(fsn ) (z1 ) − (fsn ) (z2 )| ≤ CN |z1 − z2 |,

for all s ≤ N, z ∈ RN .

(4.7)

If 0 ≤ t1 ≤ t2 ≤ N and z1 , z2 ∈ RN , equation (4.5) implies that

|ftn (z1 ) − ftn (z2 )| ≤ |z1 − z2 | + 2
2

1

+2

(fsn ) (z1 ) (fsn ) (z2 ) n
−
dµ (s, u)
z2 − u
[0,t1 ]×R z1 − u

(fsn ) (z2 ) n
dµ (s, u) = |z1 − z2 | + 2(I) + 2(II).
[t1 ,t2 ]×R z2 − u

If we can bound (I) and (II) uniformly in N , |t2 − t1 |, and |z1 − z2 |, we will have that f n is
equicontinuous in [0, N ] × RN .
For any u ∈ R and z ∈ RN , we have |z − u| ≥ 1/N . Therefore, (II) can be bounded by

(II) ≤ CN

2
dµ(s, u) ≤ CN (2N )|t2 − t1 |.
[t1 ,t2 ]×R |z − u|

To estimate (I), observe that for any u ∈ R, we have
(fsn ) (z1 ) (fsn ) (z2 )
(fsn ) (z1 ) (fsn ) (z1 )
(fsn ) (z1 ) (fsn ) (z2 )
−
≤
−
+
−
z1 − u
z2 − u
z1 − u
z2 − u
z2 − u
z2 − u

116

≤ CN

z2 − z1
+ N (fsn ) (z1 ) − (fsn ) (z2 )
(z1 − u)(z2 − u)
≤ N 2 CN |z1 − z2 | + N CN |z1 − z2 |,

where the last inequality follows from (4.7). Therefore, since the marginal of µn is Lebesgue
measure and t1 ≤ N , we have

(I) ≤
[0,t1 ]×R

N 2 CN |z1 − z2 | + N CN |z1 − z2 |dµ(s, u) ≤ t1 N 2 CN + N CN |z1 − z2 |

≤ N 3 CN + N 2 CN |z1 − z2 |.
Therefore, we have equicontinuity of f n on [0, N ] × KN for every N . By the ArzelaAscoli theorem, there is a subsequence so that f nk → f N for each N locally uniformly on
[0, N ] × KN . Taking N → ∞ and applying a diagonal argument gives the desired limit
function f : [0, ∞) × H.

Lemma 27. If (gtn ) is a solution of the chordal Loewner equation driven by µn , (gtn ) → (gt )
in G, and µn → µ in N , then (gt ) is a solution to the chordal Loewner equation driven by µ.
Proof. The chordal equation (4.1) implies that

gtn (z) =

[0,t]×R

Φ(u, gsn (z)) − Φ(u, gs (z))dµn (s, u) +

117

Φ(u, gs (z))dµn (s, u) + z. (4.8)
[0,t]×R

To estimate this, we will need to show that

lim

sup {|gsn (z) − gs (z)|} = 0.

n→∞ 0≤s≤t

(4.9)

This will be justified at the end of the proof. It is easy to show that Im(gsn (z)) is decreasing
in s for each n, so taking the limit as n → ∞ shows that Im(gs (z)) is also a decreasing
function of s. It also follows that Im(gsn (z)) ≥ min{Im(gt (z)), Im(gt1 (z)), · · · } = h > 0 for
each n ∈ N and s ≤ t. Strict inequality follows since {gtn (z)}∞
n=1 is a sequence of numbers
in H converging to gt (z) ∈ H. By Lemma 20, if L = 2/h2 ,

[0,t]×R

|Φ(u, gsn (z)) − Φ(u, gs (z))|dµn (s, u) ≤

[0,t]×R

L|gsn (z) − gs (z)|dµn (s, u)

≤ L sup{|gsn (z) − gs (z)|}µ ([0, t] × R) = Lt sup{|gsn (z) − gs (z)|} → 0
s≤t

s≤t

as n → ∞ by (4.9).
To estimate the second term in equation (4.8), observe that for a fixed z, since s → gs (z)
is continuous, the mapping (s, u) → Φ(u, gs (z)) is a continuous function. Since µn → µ
weakly on [0, t] × R, it follows that

Φ(u, gs (z))dµn (s, u) →
[0,t]×R

Φ(u, gs (z))dµ(s, u).
[0,t]×R

Therefore, the above and (4.9) imply that

gt (z) = lim gtn (z) = lim equation (4.8) =
n→∞

n→∞

118

Φ(u, gs (z))dµ(s, u) + z.
[0,t]×R

Now we prove (4.9). Suppose otherwise, so that there is some

> 0, some subsequence

nk , and some sequence sk ∈ [0, t] with

n

|gskk (z) − gsk (z)| ≥ .

(4.10)

Passing to a new subsequence if necessary, we can assume that there is some s∗ ∈ [0, t] with
sk → s∗ . Since gt (z) is continuous in t, gsk (z) converges to gs∗ (z) as k → ∞. We also claim
n

that gskk (z) is bounded. To see this, if h is defined as above,
2
dµnk (s, u)
nk
[0,sk ]×R |gs (z) − u|

n

|gskk (z) − z| ≤

≤

2t
2sk
≤ .
h
h
n

Therefore, passing to a subsequence if necessary, we can assume that gskk (z) converges to
n
some w ∈ H as k → ∞. Since ft k (z) converges to ft (z) locally uniformly in (t, z), it follows
n

n

that z = fskk (gskk (z)) → fs∗ (w), and so gs∗ (z) = w. Since gsk (z) → gs∗ (z), we get that
n

lim gskk (z) − gsk (z) = gs∗ (z) − gs∗ (z) = 0,
k→∞

which contradicts (4.10).

Finally, we can prove existence.
Proposition 9.

a) If µ ∈ N , then the solution to the chordal Loewner equation driven

by µ exists.
b) If (gt ) is the solution of the chordal Loewner equation driven by µ ∈ N , and if ft = gt−1 ,

119

then (4.5) holds for f . That is,

ft (z) − z =

−2fs (z)
dµ(s, u).
[0,t]×R z − u

c) Assume µn → µ in N , and suppose the respective corresponding Loewner maps are (gtn )
and (gt ). Assume further that the associated hulls {Ktn }n∈N are uniformly bounded for
each t > 0. Then gtn → gt in G.
n
Proof. Lemma 26 can be applied to the sequence dµn
t dt ∈ N to conclude that (gt ) → (gt )

subsequentially in G to some family of functions (gt ). By Lemma 27 it follows that (gt )
solves the chordal Loewner equation driven by µ, which proves a).
To prove b), Lemma 26 can be applied to f n (after passing to a subsequence if necessary),
to conclude that |f n (z) − f (z)| → 0. Therefore, it suffices to show that the difference of
(4.5) for ftn (z) and ft (z) goes to 0. To see this, observe that
−2fs (z)
−2(fsn ) (z) n
dµ (s, u) −
dµ(s, u)
z−u
[0,t]×R
[0,t]×R z − u

≤2

fs (z) (fsn ) (z)
−
dµn (s, u)+2
z−u
[0,t]×R z − u

fs (z)
fs (z) n
dµ(s, u) −
dµ (s, u) .
[0,t]×R z − u
[0,t]×R z − u

The first term goes to 0 by the dominated convergence theorem, since (fsn )s≤t is a normal
family, so is the family of derivatives. Therefore, |(f n )s (z)| can be bounded uniformly in
s ≤ t. Since fsn (z) → fs (z) locally uniformly in z for all s ≤ t, and so does (f n )s . The
second term goes to 0 because µn → µ.
The last assertion follows from Lemmas 25, the proof of 26 with b), and 27. In particular,
by b) and Lemma 25, the remark after Lemma 26 implies that there is a subsequential limit

120

gt for which f n converges to f uniformly in both space and time. Lemma 27 implies that
gt is a solution to the Loewner equation driven by µ. Also, Lemma 27 implies that any
subsequential limit of gtn solves the Loewner equation driven by µ, which must be unique by
Proposition 8, and therefore the sequence (gtn ) converges to (gt ).

4.6

Growing Hulls

In this section, we prove the reverse direction of Theorem 8. We start with a continuously
growing family of hulls (Kt ) and construct a measure µ ∈ N whose solution to (4.1) generates
the family (Kt ).
Proposition 10. Let Kt be growing hulls with hcapKt = 2t for every t. Then there is a
measure µ ∈ N so that gt : H\Kt → H, where gt solves the Loewner equation driven by µ.
We first prove the following lemma. The proof is identical to that of Lemma 6.5 in [19].
Lemma 28. Let K be an H-hull, and let 2T =hcap(K). Then there exists some µ ∈ NT so
that if gt is the solution to the Loewner equation driven by µ, then gT : H\K → H.
Proof. Let

> 0, and let γ : [0, T ] → H be a simple curve so that γ begins in R, encloses

ˆ less than , where K
ˆ is the union of K and the convex
K, and has Hausdorff distance from K
hull of K ∩ R. By ([7], Proposition 4.4), there exists a continuous function λ so that if gt is
the solution of the chordal Loewner equation driven by λ , then gt : H\γ [0, t] → H for all
t. If we define µ = dδλ (t) dt and take a subsequential weak limit as

→ 0, Proposition 9

implies that there is a measure µ ∈ N whose solution gt satisfies gT : H\K → H.

121

Proof of Proposition 10. Suppose for now that gt = gKt for each t. Let δ > 0, and let
l ∈ {0, 1, 2, · · · }. Then hcap(Kδ(l+1) /Kδl ) = 2δ. By Lemma 28, there exists a measure
δ,l

µ
˜δ,l on [0, δ] × R so that if g˜t

δ,l

solves the Loewner equation driven by µ
˜δ,l , then g˜δ

:

H\(Kδ(l+1) /Kδl ) → H. Define a new measure µδ,l on [δl, δ(l + 1)] × R by shifting the
previous measure. That is,

µδ,l (E) = µ
˜δ,l ({(x − δl, y) : (x, y) ∈ E}).

∞
δ,l
l=0 µ ,

Finally, define a measure µδ ∈ N by µδ =

and let gtδ solve the Loewner equation

driven by µδ with gtδ : H\Ktδ → H.
δ for every l and δ. We prove this by induction. The case
We will show that Kδl = Kδl

l = 0 is obvious, since if t ≤ δ, then

gtδ (z) = z +

[0,t]×R

Φ(u, gsδ (z))d˜
µδ,0 (s, u),

δ,0

and therefore gδδ = g˜t (z) for each t ≤ δ, and Kδ /K0 = Kδ .
δ : H → H\K . For any t ∈ [0, δ],
Suppose this is true for l. Then fδl
δl

δ,l

δ
δ (z)) = f δ (z) +
Gt (z) := gδl+t
(fδl
δl

+
[δl,δl+t]×R

δ (z)))dµδ (s, u)
Φ(u, gsδ (fδl

δ (z) + (g δ (f δ (z)) − f δ (z)) +
= fδl
δl
δl δl

=z+
[0,t]×R

[0,δl]×R

δ (z)))dµδ (s, u)
Φ(u, gsδ (fδl

[δl,δl+t]

δ (z)))dµδ,l (s, u)
Φ(u, gsδ (fδl

δ
δ (z)))d˜
Φ(u, gδl+s
(fδl
µδ,l (s, u)

122

δ,l

=z+
[0,t]×R

Φ(u, Gs (z))d˜
µδ,l (s, u).

δ,l

δ,l

Thus, Gt solves the Loewner equation driven by µ
˜δ,l , and so Gδ = g˜δδ : H\(Kδ(l+1) /Kδl ) →
H. It follows that
δ
δ =K
Kδ(l+1)
/Kδl
δ(l+1) /Kδl .
δ =g
Therefore, since gδl
Kδl by the induction hypothesis,

δ
gδ(l+1)
=g δ
K

/K δ
δ(l+1) δl

δ =g
◦ gδl
Kδ(l+1) /Kδl ◦ gKδl = gKδ(l+1) .

If µk = µδk is a sequence with a weak limit µ ∈ N , which exists because the hulls Ktδ are
uniformly bounded for each t < ∞, then Proposition 9 implies that gtk converges to gt , the
solution of the Loewner equation driven by µ. Since f k → f locally uniformly in space and
time, it follows that Ktk → Kt in the Carath´eodory topology for every t. Hence, gt solves
the Loewner equation driven by µ, and gt : H\Kt → H.

The last part of Theorem 8 which remains to be proven is the convergence fact. We
need to prove that convergence of a family of hulls implies convergence of the corresponding
measures.
Proposition 11. Suppose that µn ∈ N , and that gtn : H\Ktn → H is the associated solution
of the chordal Loewner equation. Then if Ktn → Kt in the Carath´eodory topology for some
increasing family of hulls Kt and the family {Ktn }n∈N is uniformly bounded for each t, then
there is a measure µ ∈ N so that (Kt ) are the hulls associated with µ and µn → µ in N .
Proof. By Proposition 9, we know that there is a measure µ which drives the family of

123

hulls Kt . Let µ
˜ be any subsequential limit of µn , which exists because the hulls Ktn are
n
bounded for each t < ∞. Also by Proposition 9, µnk → µ
˜ implies that Kt k converges in
n

the Carath´eodory topology to the hulls driven by µ
˜. Since we know that Kt k converges to
Kt , and the hulls are uniquely determined by the measures, it follows that µ
˜ = µ. Since µ
˜
was an arbitrary subsequential limit, it follows that µn → µ in N .

124

BIBLIOGRAPHY

125

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